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Partitionsfunktion – Wikipedia
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border-bottom-width: 1px; font-size:95%; margin-bottom:1em; padding: 0.25em; overflow: hidden; word-break: break-word; word-wrap: break-word;" id="Vorlage_Dieser_Artikel"><div class="noviewer noresize" style="display: table-cell; padding-bottom: 0.2em; padding-left: 0.25em; padding-right: 1em; padding-top: 0.2em; vertical-align: middle;" id="bksicon" aria-hidden="true" role="presentation"><span typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/e/ea/Disambig-dark.svg/25px-Disambig-dark.svg.png" decoding="async" width="25" height="19" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/ea/Disambig-dark.svg/38px-Disambig-dark.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/ea/Disambig-dark.svg/50px-Disambig-dark.svg.png 2x" data-file-width="444" data-file-height="340" /></span></span></div> <div style="display: table-cell; vertical-align: middle; width: 100%;"> <div role="navigation"> Dieser Artikel beschreibt die Partitionsfunktion in der Mathematik. Zur Partitionsfunktion in der Statistischen Physik siehe <a href="/wiki/Zustandssumme" title="Zustandssumme">Zustandssumme</a>.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup></div> </div></div> <p>Die <b>Partitionsfunktionen</b> geben die Anzahl der Möglichkeiten an, positive, <a href="/wiki/Ganze_Zahl" title="Ganze Zahl">ganze Zahlen</a> in positive, ganze <a href="/wiki/Summand" class="mw-redirect" title="Summand">Summanden</a> zu zerlegen. Üblicherweise betrachtet man die Zerlegungen ohne Berücksichtigung der Reihenfolge. Jede solche Zerlegung wird in der <a href="/wiki/Kombinatorik" title="Kombinatorik">Kombinatorik</a> als (ungeordnete) <b>Zahlpartition</b><sup id="cite_ref-MaNes_2-0" class="reference"><a href="#cite_note-MaNes-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> oder manchmal kurz <b>Partition</b><sup id="cite_ref-MaNes_2-1" class="reference"><a href="#cite_note-MaNes-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> bezeichnet. Die Bestimmung aller Zahlpartitionen für eine bestimmte (große) natürliche Zahl ist ein wichtiges Problem sowohl in der theoretischen als auch der praktischen <a href="/wiki/Informatik" title="Informatik">Informatik</a>. Siehe dazu den Artikel <a href="/wiki/Partitionierungsproblem" title="Partitionierungsproblem">Partitionierungsproblem</a>. </p><p>Die Partitionsfunktion ohne Nebenbedingungen (Anzahl der ungeordneten Zahlpartitionen von <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span>) wird als <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(n)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5e303d2c14cd399b6f52b468c9fd44a542bed422" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.949ex; height:2.843ex;" alt="{\displaystyle P(n)}"></span>, manchmal auch als <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p(n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p(n)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0e7d5ae3aa9524f57fb8b44ac46ee8cf6a52d7e6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:4.463ex; height:2.843ex;" alt="{\displaystyle p(n)}"></span> notiert und ist Folge <a href="//oeis.org/A000041" class="extiw" title="oeis:A000041">A000041</a> in <a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>. Es gibt eine Reihe von Funktionen, bei denen an die Summanden zusätzliche Bedingungen gestellt werden, zum Beispiel dass jeder Summand nur einmal vorkommen darf (<b>strikte Zahlpartitionen</b>). Diese Variante wird ebenfalls Partitionsfunktion, manchmal auch <b>strikte Partitionsfunktion</b> genannt, als <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q(n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Q</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q(n)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/39f1477b71cf9622d2d860e02dbbcffaeac2f13a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.042ex; height:2.843ex;" alt="{\displaystyle Q(n)}"></span> oder <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q(n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>q</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q(n)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1d3e070e63e1e4755b8b5f2cc03381e9186ab395" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.274ex; height:2.843ex;" alt="{\displaystyle q(n)}"></span> notiert und ist Folge <a href="//oeis.org/A000009" class="extiw" title="oeis:A000009">A000009</a> in <a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Datei:Partitionsfunktion_pn.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/de/thumb/0/0f/Partitionsfunktion_pn.png/220px-Partitionsfunktion_pn.png" decoding="async" width="220" height="165" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/de/thumb/0/0f/Partitionsfunktion_pn.png/330px-Partitionsfunktion_pn.png 1.5x, //upload.wikimedia.org/wikipedia/de/thumb/0/0f/Partitionsfunktion_pn.png/440px-Partitionsfunktion_pn.png 2x" data-file-width="640" data-file-height="480" /></a><figcaption>Partitionsfunktion P(n) in <a href="/wiki/Halblogarithmisch" class="mw-redirect" title="Halblogarithmisch">halblogarithmischer</a> Darstellung</figcaption></figure> <p>Mit einer aus der Partitionsfunktion <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(n)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5e303d2c14cd399b6f52b468c9fd44a542bed422" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.949ex; height:2.843ex;" alt="{\displaystyle P(n)}"></span> abgeleiteten zahlentheoretischen Funktion kann die Anzahl der Isomorphietypen für die <a href="/wiki/Endlich_erzeugte_abelsche_Gruppe" title="Endlich erzeugte abelsche Gruppe">endlichen abelschen Gruppen</a> angegeben werden. </p> <div id="toc" class="toc" role="navigation" aria-labelledby="mw-toc-heading"><input type="checkbox" role="button" id="toctogglecheckbox" class="toctogglecheckbox" style="display:none" /><div class="toctitle" lang="de" dir="ltr"><h2 id="mw-toc-heading">Inhaltsverzeichnis</h2><span class="toctogglespan"><label class="toctogglelabel" for="toctogglecheckbox"></label></span></div> <ul> <li class="toclevel-1 tocsection-1"><a href="#Partition"><span class="tocnumber">1</span> <span class="toctext">Partition</span></a></li> <li class="toclevel-1 tocsection-2"><a href="#Eigenschaften_von_P(n)"><span class="tocnumber">2</span> <span class="toctext">Eigenschaften von P(n)</span></a> <ul> <li class="toclevel-2 tocsection-3"><a href="#Beispielwerte"><span class="tocnumber">2.1</span> <span class="toctext">Beispielwerte</span></a></li> <li class="toclevel-2 tocsection-4"><a href="#Rekursive_Darstellung"><span class="tocnumber">2.2</span> <span class="toctext">Rekursive Darstellung</span></a></li> <li class="toclevel-2 tocsection-5"><a href="#Asymptotisches_Verhalten"><span class="tocnumber">2.3</span> <span class="toctext">Asymptotisches Verhalten</span></a></li> <li class="toclevel-2 tocsection-6"><a href="#Erzeugende_Funktion"><span class="tocnumber">2.4</span> <span class="toctext">Erzeugende Funktion</span></a> <ul> <li class="toclevel-3 tocsection-7"><a href="#Zusammenhang_mit_den_Pentagonalzahlen"><span class="tocnumber">2.4.1</span> <span class="toctext">Zusammenhang mit den Pentagonalzahlen</span></a></li> <li class="toclevel-3 tocsection-8"><a href="#Rekursionsformel_aus_dem_Pentagonalzahlensatz"><span class="tocnumber">2.4.2</span> <span class="toctext">Rekursionsformel aus dem Pentagonalzahlensatz</span></a></li> </ul> </li> <li class="toclevel-2 tocsection-9"><a href="#Berechnung_mit_analytischer_Zahlentheorie"><span class="tocnumber">2.5</span> <span class="toctext">Berechnung mit analytischer Zahlentheorie</span></a></li> <li class="toclevel-2 tocsection-10"><a href="#Berechnung_mit_algebraischer_Zahlentheorie"><span class="tocnumber">2.6</span> <span class="toctext">Berechnung mit algebraischer Zahlentheorie</span></a></li> <li class="toclevel-2 tocsection-11"><a href="#Kongruenzen"><span class="tocnumber">2.7</span> <span class="toctext">Kongruenzen</span></a></li> </ul> </li> <li class="toclevel-1 tocsection-12"><a href="#Ferrers-Diagramme"><span class="tocnumber">3</span> <span class="toctext">Ferrers-Diagramme</span></a> <ul> <li class="toclevel-2 tocsection-13"><a href="#Konjugierte_Partition"><span class="tocnumber">3.1</span> <span class="toctext">Konjugierte Partition</span></a></li> <li class="toclevel-2 tocsection-14"><a href="#Formalisierung"><span class="tocnumber">3.2</span> <span class="toctext">Formalisierung</span></a></li> </ul> </li> <li class="toclevel-1 tocsection-15"><a href="#Varianten"><span class="tocnumber">4</span> <span class="toctext">Varianten</span></a> <ul> <li class="toclevel-2 tocsection-16"><a href="#Partitionen_mit_vorgegebenem_kleinstem_Summanden,_p(k,n)"><span class="tocnumber">4.1</span> <span class="toctext">Partitionen mit vorgegebenem kleinstem Summanden, p(k,n)</span></a> <ul> <li class="toclevel-3 tocsection-17"><a href="#Rekursionsformel_für_p(k,n)_und_P(n)"><span class="tocnumber">4.1.1</span> <span class="toctext">Rekursionsformel für p(k,n) und P(n)</span></a></li> </ul> </li> <li class="toclevel-2 tocsection-18"><a href="#Geordnete_Zahlpartitionen"><span class="tocnumber">4.2</span> <span class="toctext">Geordnete Zahlpartitionen</span></a></li> <li class="toclevel-2 tocsection-19"><a href="#Strikte_Partitionen_und_verwandte_Nebenbedingungen"><span class="tocnumber">4.3</span> <span class="toctext">Strikte Partitionen und verwandte Nebenbedingungen</span></a></li> </ul> </li> <li class="toclevel-1 tocsection-20"><a href="#Mathematische_Anwendungen"><span class="tocnumber">5</span> <span class="toctext">Mathematische Anwendungen</span></a> <ul> <li class="toclevel-2 tocsection-21"><a href="#Konjugationsklassen_der_symmetrischen_Gruppe"><span class="tocnumber">5.1</span> <span class="toctext">Konjugationsklassen der symmetrischen Gruppe</span></a></li> <li class="toclevel-2 tocsection-22"><a href="#Zahlpartition_und_endliche_Mengenpartition"><span class="tocnumber">5.2</span> <span class="toctext">Zahlpartition und endliche Mengenpartition</span></a></li> <li class="toclevel-2 tocsection-23"><a href="#Endliche_abelsche_p-Gruppen_und_abelsche_Gruppen"><span class="tocnumber">5.3</span> <span class="toctext">Endliche abelsche p-Gruppen und abelsche Gruppen</span></a> <ul> <li class="toclevel-3 tocsection-24"><a href="#Anzahlfunktion_von_Isomorphietypen_endlicher_abelscher_Gruppen"><span class="tocnumber">5.3.1</span> <span class="toctext">Anzahlfunktion von Isomorphietypen endlicher abelscher Gruppen</span></a></li> </ul> </li> </ul> </li> <li class="toclevel-1 tocsection-25"><a href="#Strikte_Partitionsfunktion"><span class="tocnumber">6</span> <span class="toctext">Strikte Partitionsfunktion</span></a> <ul> <li class="toclevel-2 tocsection-26"><a href="#Definition_und_Eigenschaften_der_strikten_Partitionen"><span class="tocnumber">6.1</span> <span class="toctext">Definition und Eigenschaften der strikten Partitionen</span></a></li> <li class="toclevel-2 tocsection-27"><a href="#Beispielwerte_der_strikten_Partitionszahlen"><span class="tocnumber">6.2</span> <span class="toctext">Beispielwerte der strikten Partitionszahlen</span></a></li> <li class="toclevel-2 tocsection-28"><a href="#Maclaurinsche_Reihe_der_strikten_Partitionszahlen"><span class="tocnumber">6.3</span> <span class="toctext">Maclaurinsche Reihe der strikten Partitionszahlen</span></a></li> <li class="toclevel-2 tocsection-29"><a href="#Identitäten_über_die_strikte_Partitionsfunktion"><span class="tocnumber">6.4</span> <span class="toctext">Identitäten über die strikte Partitionsfunktion</span></a></li> </ul> </li> <li class="toclevel-1 tocsection-30"><a href="#Oberpartitionsfunktion"><span class="tocnumber">7</span> <span class="toctext">Oberpartitionsfunktion</span></a> <ul> <li class="toclevel-2 tocsection-31"><a href="#Definition_der_Oberpartitionen"><span class="tocnumber">7.1</span> <span class="toctext">Definition der Oberpartitionen</span></a></li> <li class="toclevel-2 tocsection-32"><a href="#Beispielwerte_der_Oberpartitionszahlen"><span class="tocnumber">7.2</span> <span class="toctext">Beispielwerte der Oberpartitionszahlen</span></a></li> <li class="toclevel-2 tocsection-33"><a href="#Maclaurinsche_Reihe_der_Oberpartitionszahlen"><span class="tocnumber">7.3</span> <span class="toctext">Maclaurinsche Reihe der Oberpartitionszahlen</span></a></li> <li class="toclevel-2 tocsection-34"><a href="#Identitäten_über_die_Oberpartitionsfunktion"><span class="tocnumber">7.4</span> <span class="toctext">Identitäten über die Oberpartitionsfunktion</span></a></li> </ul> </li> <li class="toclevel-1 tocsection-35"><a href="#Literatur"><span class="tocnumber">8</span> <span class="toctext">Literatur</span></a></li> <li class="toclevel-1 tocsection-36"><a href="#Weblinks"><span class="tocnumber">9</span> <span class="toctext">Weblinks</span></a></li> <li class="toclevel-1 tocsection-37"><a href="#Einzelnachweise"><span class="tocnumber">10</span> <span class="toctext">Einzelnachweise</span></a></li> </ul> </div> <div class="mw-heading mw-heading2"><h2 id="Partition">Partition</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Partitionsfunktion&veaction=edit&section=1" title="Abschnitt bearbeiten: Partition" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Partitionsfunktion&action=edit&section=1" title="Quellcode des Abschnitts bearbeiten: Partition"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Eine <b>Partition</b> ist eine (endliche oder unendliche) <a href="/wiki/Folge_(Mathematik)" title="Folge (Mathematik)">Folge</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \kappa =(k_{1},k_{2},\dots ,k_{p},\dots )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>κ<!-- κ --></mi> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>…<!-- … --></mo> <mo>,</mo> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo>,</mo> <mo>…<!-- … --></mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \kappa =(k_{1},k_{2},\dots ,k_{p},\dots )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/42e74175ac855b4c9a1d2dd0c20ae1de56eed2b6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:23.017ex; height:3.009ex;" alt="{\displaystyle \kappa =(k_{1},k_{2},\dots ,k_{p},\dots )}"></span> bestehend aus nicht-negativen ganzen Zahlen </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k_{1}\geq k_{2}\geq \cdots \geq k_{p}\geq \cdots \geq 0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>≥<!-- ≥ --></mo> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>≥<!-- ≥ --></mo> <mo>⋯<!-- ⋯ --></mo> <mo>≥<!-- ≥ --></mo> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo>≥<!-- ≥ --></mo> <mo>⋯<!-- ⋯ --></mo> <mo>≥<!-- ≥ --></mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k_{1}\geq k_{2}\geq \cdots \geq k_{p}\geq \cdots \geq 0.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f965ea081069322889edea4fc62de72dba579ef8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:29.549ex; height:2.843ex;" alt="{\displaystyle k_{1}\geq k_{2}\geq \cdots \geq k_{p}\geq \cdots \geq 0.}"></span></dd></dl> <p>Die Zahl <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\kappa |=k_{1}+k_{2}+\cdots +k_{p}+\cdots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>κ<!-- κ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>=</mo> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <mo>⋯<!-- ⋯ --></mo> <mo>+</mo> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo>+</mo> <mo>⋯<!-- ⋯ --></mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |\kappa |=k_{1}+k_{2}+\cdots +k_{p}+\cdots }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2102aefb33c442de4d8ef19ae1df599b98163cd1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:29.341ex; height:3.009ex;" alt="{\displaystyle |\kappa |=k_{1}+k_{2}+\cdots +k_{p}+\cdots }"></span> nennt man <i>Gewicht</i>. Die Folgenglieder, welche nicht Null sind <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k_{i}\neq 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>≠<!-- ≠ --></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k_{i}\neq 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2f4931dc90b45f8f4b6096525216bd77b94c9123" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.272ex; height:2.676ex;" alt="{\displaystyle k_{i}\neq 0}"></span>, nennt man <i>Teile</i> von <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \kappa }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>κ<!-- κ --></mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \kappa }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/54ddec2e922c5caea4e47d04feef86e782dc8e6d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.339ex; height:1.676ex;" alt="{\displaystyle \kappa }"></span> und die Anzahl der Teile <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle l(\kappa )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>l</mi> <mo stretchy="false">(</mo> <mi>κ<!-- κ --></mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle l(\kappa )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4cd146be4f4c139a336a33414e27a9253ae8fc62" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.842ex; height:2.843ex;" alt="{\displaystyle l(\kappa )}"></span> nennt man <i>Länge</i> der Partition. Partitionen, welche die gleichen Teile haben, werden wir miteinander identifizieren, das heißt </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \kappa _{1}=(2,1,0,0,\dots ),\quad \kappa _{2}=(2,1,0),\quad \kappa _{3}=(2,1),\quad }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>κ<!-- κ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>0</mn> <mo>,</mo> <mn>0</mn> <mo>,</mo> <mo>…<!-- … --></mo> <mo stretchy="false">)</mo> <mo>,</mo> <mspace width="1em" /> <msub> <mi>κ<!-- κ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>,</mo> <mspace width="1em" /> <msub> <mi>κ<!-- κ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>=</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>,</mo> <mspace width="1em" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \kappa _{1}=(2,1,0,0,\dots ),\quad \kappa _{2}=(2,1,0),\quad \kappa _{3}=(2,1),\quad }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81e4ec6c89be6dab3cf89604923c70b9e9db1909" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:52.008ex; height:2.843ex;" alt="{\displaystyle \kappa _{1}=(2,1,0,0,\dots ),\quad \kappa _{2}=(2,1,0),\quad \kappa _{3}=(2,1),\quad }"></span></dd></dl> <p>beschreiben dieselbe Partition mit Länge <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle l(\kappa _{1})=l(\kappa _{2})=l(\kappa _{3})=2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>l</mi> <mo stretchy="false">(</mo> <msub> <mi>κ<!-- κ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mi>l</mi> <mo stretchy="false">(</mo> <msub> <mi>κ<!-- κ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mi>l</mi> <mo stretchy="false">(</mo> <msub> <mi>κ<!-- κ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle l(\kappa _{1})=l(\kappa _{2})=l(\kappa _{3})=2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ae7a54d9bd15d77037d73af6dd20d07a2f14a3e5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.145ex; height:2.843ex;" alt="{\displaystyle l(\kappa _{1})=l(\kappa _{2})=l(\kappa _{3})=2}"></span>. </p><p>Wenn <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n=|\kappa |}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>κ<!-- κ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n=|\kappa |}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/482009ba72ea5a2bb285efc64f6b3fdbe26f5445" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.126ex; height:2.843ex;" alt="{\displaystyle n=|\kappa |}"></span> gilt, dann ist <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \kappa }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>κ<!-- κ --></mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \kappa }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/54ddec2e922c5caea4e47d04feef86e782dc8e6d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.339ex; height:1.676ex;" alt="{\displaystyle \kappa }"></span> eine <i>Partition von <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span></i>.<sup id="cite_ref-Macdonald_4-0" class="reference"><a href="#cite_note-Macdonald-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Eigenschaften_von_P(n)"><span id="Eigenschaften_von_P.28n.29"></span>Eigenschaften von P(n)</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Partitionsfunktion&veaction=edit&section=2" title="Abschnitt bearbeiten: Eigenschaften von P(n)" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Partitionsfunktion&action=edit&section=2" title="Quellcode des Abschnitts bearbeiten: Eigenschaften von P(n)"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Beispielwerte">Beispielwerte</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Partitionsfunktion&veaction=edit&section=3" title="Abschnitt bearbeiten: Beispielwerte" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Partitionsfunktion&action=edit&section=3" title="Quellcode des Abschnitts bearbeiten: Beispielwerte"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <table class="wikitable"> <caption>Beispielwerte von P(n) und zugehörige Zahlpartitionen </caption> <tbody><tr> <th>n</th> <th>P(n)</th> <th>Zahlpartitionen </th></tr> <tr> <td>0</td> <td>1</td> <td>() leere Partition/<a href="/wiki/Leere_Summe" title="Leere Summe">leere Summe</a> </td></tr> <tr> <td>1</td> <td>1</td> <td>(1) </td></tr> <tr> <td>2</td> <td>2</td> <td>(1+1), (2) </td></tr> <tr> <td>3</td> <td>3</td> <td>(1+1+1), (1+2), (3) </td></tr> <tr> <td>4</td> <td>5</td> <td>(1+1+1+1), (1+1+2), (2+2), (1+3), (4) </td></tr> <tr> <td>5</td> <td>7</td> <td>(1+1+1+1+1), (1+1+1+2), (1+2+2), (1+1+3), (2+3), (1+4), (5) </td></tr> <tr> <td>6</td> <td>11</td> <td>(1+1+1+1+1+1), (1+1+1+1+2), (1+1+2+2), (2+2+2), (1+1+1+3), (1+2+3), (3+3), (1+1+4), (2+4), (1+5), (6) </td></tr></tbody></table> <p>Die Werte steigen danach schnell an (siehe Folge <a href="//oeis.org/A000041" class="extiw" title="oeis:A000041">A000041</a> in <a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>): </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{array}{lcr|lcr|lcr}P(7)&=&15&P(45)&=&89\,134\\P(8)&=&22&P(100)&=&190\,569\,292&\\P(9)&=&30&P(200)&\approx &3{,}973\cdot 10^{12}\\P(10)&=&42&P(1000)&\approx &2{,}406\cdot 10^{31}\end{array}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="left center right left center right left center right" rowspacing="4pt" columnspacing="1em" columnlines="none none solid none none solid none none"> <mtr> <mtd> <mi>P</mi> <mo stretchy="false">(</mo> <mn>7</mn> <mo stretchy="false">)</mo> </mtd> <mtd> <mo>=</mo> </mtd> <mtd> <mn>15</mn> </mtd> <mtd> <mi>P</mi> <mo stretchy="false">(</mo> <mn>45</mn> <mo stretchy="false">)</mo> </mtd> <mtd> <mo>=</mo> </mtd> <mtd> <mn>89</mn> <mspace width="thinmathspace" /> <mn>134</mn> </mtd> </mtr> <mtr> <mtd> <mi>P</mi> <mo stretchy="false">(</mo> <mn>8</mn> <mo stretchy="false">)</mo> </mtd> <mtd> <mo>=</mo> </mtd> <mtd> <mn>22</mn> </mtd> <mtd> <mi>P</mi> <mo stretchy="false">(</mo> <mn>100</mn> <mo stretchy="false">)</mo> </mtd> <mtd> <mo>=</mo> </mtd> <mtd> <mn>190</mn> <mspace width="thinmathspace" /> <mn>569</mn> <mspace width="thinmathspace" /> <mn>292</mn> </mtd> <mtd /> </mtr> <mtr> <mtd> <mi>P</mi> <mo stretchy="false">(</mo> <mn>9</mn> <mo stretchy="false">)</mo> </mtd> <mtd> <mo>=</mo> </mtd> <mtd> <mn>30</mn> </mtd> <mtd> <mi>P</mi> <mo stretchy="false">(</mo> <mn>200</mn> <mo stretchy="false">)</mo> </mtd> <mtd> <mo>≈<!-- ≈ --></mo> </mtd> <mtd> <mn>3,973</mn> <mo>⋅<!-- ⋅ --></mo> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> </msup> </mtd> </mtr> <mtr> <mtd> <mi>P</mi> <mo stretchy="false">(</mo> <mn>10</mn> <mo stretchy="false">)</mo> </mtd> <mtd> <mo>=</mo> </mtd> <mtd> <mn>42</mn> </mtd> <mtd> <mi>P</mi> <mo stretchy="false">(</mo> <mn>1000</mn> <mo stretchy="false">)</mo> </mtd> <mtd> <mo>≈<!-- ≈ --></mo> </mtd> <mtd> <mn>2,406</mn> <mo>⋅<!-- ⋅ --></mo> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>31</mn> </mrow> </msup> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{array}{lcr|lcr|lcr}P(7)&=&15&P(45)&=&89\,134\\P(8)&=&22&P(100)&=&190\,569\,292&\\P(9)&=&30&P(200)&\approx &3{,}973\cdot 10^{12}\\P(10)&=&42&P(1000)&\approx &2{,}406\cdot 10^{31}\end{array}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/db77e3d9661f37935b85fed7a9fc10261a4f46dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.671ex; width:47.807ex; height:14.509ex;" alt="{\displaystyle {\begin{array}{lcr|lcr|lcr}P(7)&=&15&P(45)&=&89\,134\\P(8)&=&22&P(100)&=&190\,569\,292&\\P(9)&=&30&P(200)&\approx &3{,}973\cdot 10^{12}\\P(10)&=&42&P(1000)&\approx &2{,}406\cdot 10^{31}\end{array}}}"></span></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Rekursive_Darstellung">Rekursive Darstellung</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Partitionsfunktion&veaction=edit&section=4" title="Abschnitt bearbeiten: Rekursive Darstellung" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Partitionsfunktion&action=edit&section=4" title="Quellcode des Abschnitts bearbeiten: Rekursive Darstellung"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <table class="wikitable float-right"> <caption>Beispielwerte von P(n,k) </caption> <tbody><tr class="hintergrundfarbe5"> <td colspan="2" rowspan="2"><b>P(n,k)</b></td> <td colspan="10" style="text-align:center"><b>k</b> </td></tr> <tr align="right"> <th>1</th> <th>2</th> <th>3</th> <th>4</th> <th>5</th> <th>6</th> <th>7</th> <th>8</th> <th>9</th> <th>10 </th></tr> <tr align="right"> <td rowspan="10" class="hintergrundfarbe5" align="center"><b>n</b></td> <td><b>1</b></td> <td>1 </td></tr> <tr align="right"> <td><b>2</b></td> <td>1</td> <td>1 </td></tr> <tr align="right"> <td><b>3</b></td> <td>1</td> <td>1</td> <td>1 </td></tr> <tr align="right"> <td><b>4</b></td> <td>1</td> <td>2</td> <td>1</td> <td>1 </td></tr> <tr align="right"> <td><b>5</b></td> <td>1</td> <td>2</td> <td>2</td> <td>1</td> <td>1 </td></tr> <tr align="right"> <td><b>6</b></td> <td>1</td> <td>3</td> <td>3</td> <td>2</td> <td>1</td> <td>1 </td></tr> <tr align="right"> <td><b>7</b></td> <td>1</td> <td>3</td> <td>4</td> <td>3</td> <td>2</td> <td>1</td> <td>1 </td></tr> <tr align="right"> <td><b>8</b></td> <td>1</td> <td>4</td> <td>5</td> <td>5</td> <td>3</td> <td>2</td> <td>1</td> <td>1 </td></tr> <tr align="right"> <td><b>9</b></td> <td>1</td> <td>4</td> <td>7</td> <td>6</td> <td>5</td> <td>3</td> <td>2</td> <td>1</td> <td>1 </td></tr> <tr align="right"> <td><b>10</b></td> <td>1</td> <td>5</td> <td>8</td> <td>9</td> <td>7</td> <td>5</td> <td>3</td> <td>2</td> <td>1</td> <td>1 </td></tr></tbody></table> <p>Bezeichnet <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(n,k)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>,</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(n,k)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7129f7bbca422c232ee252195cab5568be1cdc99" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.195ex; height:2.843ex;" alt="{\displaystyle P(n,k)}"></span> die Anzahl der Möglichkeiten, die positive, ganze Zahl <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span> in genau <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}"></span> positive, ganze Summanden zu zerlegen, dann gilt </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(n)=\sum _{i=1}^{n}P(n,i)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munderover> <mo>∑<!-- ∑ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mi>P</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>,</mo> <mi>i</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(n)=\sum _{i=1}^{n}P(n,i)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b9a77ca9960850fe1ae94cbd2f6e6c1dfe86e0d0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:18.576ex; height:6.843ex;" alt="{\displaystyle P(n)=\sum _{i=1}^{n}P(n,i)}"></span>,</dd></dl> <p>wobei sich die Zahlen <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(n,k)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>,</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(n,k)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7129f7bbca422c232ee252195cab5568be1cdc99" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.195ex; height:2.843ex;" alt="{\displaystyle P(n,k)}"></span> rekursiv über <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(n,1)=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(n,1)=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/da64c3c6e400d10322fddcd4c6b679114134bd34" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.407ex; height:2.843ex;" alt="{\displaystyle P(n,1)=1}"></span> und <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(n,n)=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>,</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(n,n)=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0259eaeb473acae2cd346d5799c1eb97d34acd75" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.639ex; height:2.843ex;" alt="{\displaystyle P(n,n)=1}"></span> sowie </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(n+k,k)=\sum _{j=1}^{k}P(n,j)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mi>k</mi> <mo>,</mo> <mi>k</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munderover> <mo>∑<!-- ∑ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </munderover> <mi>P</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>,</mo> <mi>j</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(n+k,k)=\sum _{j=1}^{k}P(n,j)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5f74974ad8cf606922c5d8ed2ad31d73172f9007" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:25.028ex; height:7.676ex;" alt="{\displaystyle P(n+k,k)=\sum _{j=1}^{k}P(n,j)}"></span></dd></dl> <p>oder direkt durch </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(n,k)=P(n-k,k)+P(n-1,k-1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>,</mo> <mi>k</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>P</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−<!-- − --></mo> <mi>k</mi> <mo>,</mo> <mi>k</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>P</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−<!-- − --></mo> <mn>1</mn> <mo>,</mo> <mi>k</mi> <mo>−<!-- − --></mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(n,k)=P(n-k,k)+P(n-1,k-1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/74ec70e02a9562c19361019dca37533038035237" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:39.58ex; height:2.843ex;" alt="{\displaystyle P(n,k)=P(n-k,k)+P(n-1,k-1)}"></span></dd></dl> <p>ermitteln lassen.<sup id="cite_ref-Steger_5-0" class="reference"><a href="#cite_note-Steger-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Zimmermann_6-0" class="reference"><a href="#cite_note-Zimmermann-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Asymptotisches_Verhalten">Asymptotisches Verhalten</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Partitionsfunktion&veaction=edit&section=5" title="Abschnitt bearbeiten: Asymptotisches Verhalten" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Partitionsfunktion&action=edit&section=5" title="Quellcode des Abschnitts bearbeiten: Asymptotisches Verhalten"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <table class="wikitable float-right"> <caption>Relativer Fehler der Approximationsfunktion </caption> <tbody><tr> <th><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span></th> <th>5</th> <th>10</th> <th>100</th> <th>250</th> <th>500 </th></tr> <tr> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {E(n)-P(n)}{P(n)}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>E</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>−<!-- − --></mo> <mi>P</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>P</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {E(n)-P(n)}{P(n)}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fa24ef740d9b75d72bbb8c1a7d5f7f5ac5bb530a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:13.606ex; height:6.509ex;" alt="{\displaystyle {\frac {E(n)-P(n)}{P(n)}}}"></span> in %</td> <td>27,7</td> <td>14,5</td> <td>4,57</td> <td>2,86</td> <td>2,01 </td></tr></tbody></table> <p>Für große Werte von <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span> gibt die Formel von <a href="/wiki/Godfrey_Harold_Hardy" title="Godfrey Harold Hardy">Godfrey Harold Hardy</a> und <a href="/wiki/S._Ramanujan" class="mw-redirect" title="S. Ramanujan">S. Ramanujan</a><sup id="cite_ref-MaNes_2-2" class="reference"><a href="#cite_note-MaNes-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(n)\sim E(n)={\frac {\exp \left(\pi {\sqrt {2n/3}}\right)}{4n{\sqrt {3}}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>∼<!-- ∼ --></mo> <mi>E</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>exp</mi> <mo>⁡<!-- --></mo> <mrow> <mo>(</mo> <mrow> <mi>π<!-- π --></mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>3</mn> </msqrt> </mrow> </mrow> <mo>)</mo> </mrow> </mrow> <mrow> <mn>4</mn> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>3</mn> </msqrt> </mrow> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(n)\sim E(n)={\frac {\exp \left(\pi {\sqrt {2n/3}}\right)}{4n{\sqrt {3}}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b01a5ae0da183c37bf871cadf80b7d68ee4836a5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:31.182ex; height:7.176ex;" alt="{\displaystyle P(n)\sim E(n)={\frac {\exp \left(\pi {\sqrt {2n/3}}\right)}{4n{\sqrt {3}}}}}"></span></dd></dl> <p>einen guten Näherungswert für <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(n)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5e303d2c14cd399b6f52b468c9fd44a542bed422" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.949ex; height:2.843ex;" alt="{\displaystyle P(n)}"></span>. Insbesondere bedeutet dies, dass die Anzahl der Dezimalstellen von <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(n)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5e303d2c14cd399b6f52b468c9fd44a542bed422" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.949ex; height:2.843ex;" alt="{\displaystyle P(n)}"></span> etwa proportional zur Quadratwurzel aus <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span> ist: <i>P</i>(100) hat 9 Stellen (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {100}}=10}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>100</mn> </msqrt> </mrow> <mo>=</mo> <mn>10</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {100}}=10}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a3261170b61076dd988956bf983fc7a88fef29a0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.847ex; height:2.843ex;" alt="{\displaystyle {\sqrt {100}}=10}"></span>), <i>P</i>(1000) hat 32 Stellen (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {1000}}\approx 31{,}6}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1000</mn> </msqrt> </mrow> <mo>≈<!-- ≈ --></mo> <mn>31</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>,</mo> </mrow> <mn>6</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {1000}}\approx 31{,}6}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c4052333f3b4276a0eab4f203f95eaa59be0d614" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.818ex; height:2.843ex;" alt="{\displaystyle {\sqrt {1000}}\approx 31{,}6}"></span>). </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(4n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <mn>4</mn> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(4n)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6c2f6776e96e57166156f1b76df2cbd597d3536b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.112ex; height:2.843ex;" alt="{\displaystyle P(4n)}"></span> hat etwa doppelt so viele Stellen wie <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(n)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5e303d2c14cd399b6f52b468c9fd44a542bed422" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.949ex; height:2.843ex;" alt="{\displaystyle P(n)}"></span>. </p><p>Deswegen gilt dieser Grenzwert des Quotienten sukzessiver Folgenglieder: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lim _{n\rightarrow \infty }{\frac {P(n+1)}{P(n)}}=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo stretchy="false">→<!-- → --></mo> <mi mathvariant="normal">∞<!-- ∞ --></mi> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>P</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>P</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lim _{n\rightarrow \infty }{\frac {P(n+1)}{P(n)}}=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/081493a1505fb8264f7b6b7d844a1dc813c40dd4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:18.709ex; height:6.509ex;" alt="{\displaystyle \lim _{n\rightarrow \infty }{\frac {P(n+1)}{P(n)}}=1}"></span></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Erzeugende_Funktion">Erzeugende Funktion</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Partitionsfunktion&veaction=edit&section=6" title="Abschnitt bearbeiten: Erzeugende Funktion" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Partitionsfunktion&action=edit&section=6" title="Quellcode des Abschnitts bearbeiten: Erzeugende Funktion"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Eine einfache <a href="/wiki/Erzeugende_Funktion" title="Erzeugende Funktion">erzeugende Funktion</a> für die Partitionsfunktion gewinnt man aus der multiplikativ Inversen von Eulers Funktion: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{n=0}^{\infty }P(n)x^{n}=\prod _{k=1}^{\infty }{\biggl [}\sum _{n=0}^{\infty }x^{kn}{\biggr ]}=\prod _{k=1}^{\infty }{\frac {1}{1-x^{k}}}=(x;x)_{\infty }^{-1}={\bigl [}\psi _{R}(x^{2})\vartheta _{00}(x)\vartheta _{01}(x)^{4}{\bigr ]}^{-1/6}=}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∑<!-- ∑ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞<!-- ∞ --></mi> </mrow> </munderover> <mi>P</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>=</mo> <munderover> <mo>∏<!-- ∏ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞<!-- ∞ --></mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">[</mo> </mrow> </mrow> <munderover> <mo>∑<!-- ∑ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞<!-- ∞ --></mi> </mrow> </munderover> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mi>n</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">]</mo> </mrow> </mrow> <mo>=</mo> <munderover> <mo>∏<!-- ∏ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞<!-- ∞ --></mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>1</mn> <mo>−<!-- − --></mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>=</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>;</mo> <mi>x</mi> <msubsup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞<!-- ∞ --></mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−<!-- − --></mo> <mn>1</mn> </mrow> </msubsup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">[</mo> </mrow> </mrow> <msub> <mi>ψ<!-- ψ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>R</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> <msub> <mi>ϑ<!-- ϑ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>00</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <msub> <mi>ϑ<!-- ϑ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>01</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">]</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−<!-- − --></mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>6</mn> </mrow> </msup> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{n=0}^{\infty }P(n)x^{n}=\prod _{k=1}^{\infty }{\biggl [}\sum _{n=0}^{\infty }x^{kn}{\biggr ]}=\prod _{k=1}^{\infty }{\frac {1}{1-x^{k}}}=(x;x)_{\infty }^{-1}={\bigl [}\psi _{R}(x^{2})\vartheta _{00}(x)\vartheta _{01}(x)^{4}{\bigr ]}^{-1/6}=}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ecf3232d494a4ec96b4f3a46b3715bb907f0465c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:84.816ex; height:6.843ex;" alt="{\displaystyle \sum _{n=0}^{\infty }P(n)x^{n}=\prod _{k=1}^{\infty }{\biggl [}\sum _{n=0}^{\infty }x^{kn}{\biggr ]}=\prod _{k=1}^{\infty }{\frac {1}{1-x^{k}}}=(x;x)_{\infty }^{-1}={\bigl [}\psi _{R}(x^{2})\vartheta _{00}(x)\vartheta _{01}(x)^{4}{\bigr ]}^{-1/6}=}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ={\sqrt[{6}]{2}}\,x^{1/24}\vartheta _{10}(x)^{-1/6}\vartheta _{00}(x)^{-1/6}\vartheta _{01}(x)^{-2/3}={\sqrt {3}}\,x^{1/24}\vartheta _{10}({\tfrac {1}{6}}\pi ;x^{1/6})^{-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mroot> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </mroot> </mrow> <mspace width="thinmathspace" /> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>24</mn> </mrow> </msup> <msub> <mi>ϑ<!-- ϑ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>10</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−<!-- − --></mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>6</mn> </mrow> </msup> <msub> <mi>ϑ<!-- ϑ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>00</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−<!-- − --></mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>6</mn> </mrow> </msup> <msub> <mi>ϑ<!-- ϑ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>01</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−<!-- − --></mo> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>3</mn> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>3</mn> </msqrt> </mrow> <mspace width="thinmathspace" /> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>24</mn> </mrow> </msup> <msub> <mi>ϑ<!-- ϑ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>10</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>6</mn> </mfrac> </mstyle> </mrow> <mi>π<!-- π --></mi> <mo>;</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>6</mn> </mrow> </msup> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−<!-- − --></mo> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ={\sqrt[{6}]{2}}\,x^{1/24}\vartheta _{10}(x)^{-1/6}\vartheta _{00}(x)^{-1/6}\vartheta _{01}(x)^{-2/3}={\sqrt {3}}\,x^{1/24}\vartheta _{10}({\tfrac {1}{6}}\pi ;x^{1/6})^{-1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/71ab7dca8b24e8d3cee6a3ed10e10c976948f2a0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:68.764ex; height:3.843ex;" alt="{\displaystyle ={\sqrt[{6}]{2}}\,x^{1/24}\vartheta _{10}(x)^{-1/6}\vartheta _{00}(x)^{-1/6}\vartheta _{01}(x)^{-2/3}={\sqrt {3}}\,x^{1/24}\vartheta _{10}({\tfrac {1}{6}}\pi ;x^{1/6})^{-1}}"></span></dd></dl> <p>Dabei wird mit dem Funktionskürzel <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi _{R}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ψ<!-- ψ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>R</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi _{R}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eacf8c5f2587de7c59bbfd798e11bdbeaa01707b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.993ex; height:2.509ex;" alt="{\displaystyle \psi _{R}}"></span> die <a href="/wiki/Ramanujansche_Psifunktion" title="Ramanujansche Psifunktion">Ramanujansche Psifunktion</a> zum Ausdruck gebracht. Der runde Klammerausdruck mit dem Unendlichkeitsindex stellt das <a href="/wiki/Pochhammer-Symbol" title="Pochhammer-Symbol">Pochhammer-Symbol</a> und ϑ₁₀, ϑ₀₀ und ϑ₀₁ stellen die drei <a href="/wiki/Thetafunktion" class="mw-redirect" title="Thetafunktion">Thetafunktionen</a> dar. Für das Intervall −1 ≤ x < 1 gelten alle Stellen in der ersten Zeile der gezeigten Gleichungskette. Mit dem Übergang von der zweiten Stelle zur dritten Stelle wird die Identität der <a href="/wiki/Geometrische_Reihe" title="Geometrische Reihe">geometrischen Reihe</a> dargestellt. Die restlichen gezeigten Elemente der Gleichungskette gehen unter anderem aus dem Werk <i>Evolutio producti infiniti in seriem simplicem</i> von <a href="/wiki/Leonhard_Euler" title="Leonhard Euler">Leonhard Euler</a>, aus dem Werk <i>Modular Equations and Approximations to π</i> von <a href="/wiki/Srinivasa_Ramanujan" title="Srinivasa Ramanujan">Srinivasa Ramanujan</a> und aus den Werken von <a href="/wiki/Richard_Dedekind" title="Richard Dedekind">Richard Dedekind</a> hervor. </p><p>Man erhält diese Reihe: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)=(x;x)_{\infty }^{-1}={\frac {1}{\prod _{k=1}^{\infty }(1-x^{k})}}=1+1x+2x^{2}+3x^{3}+5x^{4}+...}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>;</mo> <mi>x</mi> <msubsup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞<!-- ∞ --></mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−<!-- − --></mo> <mn>1</mn> </mrow> </msubsup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <munderover> <mo>∏<!-- ∏ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞<!-- ∞ --></mi> </mrow> </munderover> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−<!-- − --></mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>=</mo> <mn>1</mn> <mo>+</mo> <mn>1</mn> <mi>x</mi> <mo>+</mo> <mn>2</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>3</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>+</mo> <mn>5</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>+</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)=(x;x)_{\infty }^{-1}={\frac {1}{\prod _{k=1}^{\infty }(1-x^{k})}}=1+1x+2x^{2}+3x^{3}+5x^{4}+...}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/66c364a8012ec5b2c9daaeeaca61de5696b6c0f4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:66.176ex; height:6.176ex;" alt="{\displaystyle f(x)=(x;x)_{\infty }^{-1}={\frac {1}{\prod _{k=1}^{\infty }(1-x^{k})}}=1+1x+2x^{2}+3x^{3}+5x^{4}+...}"></span></dd></dl> <p>d. h., dass die Koeffizienten der Reihendarstellung von <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/202945cce41ecebb6f643f31d119c514bec7a074" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.418ex; height:2.843ex;" alt="{\displaystyle f(x)}"></span> den Werten von <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(n)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5e303d2c14cd399b6f52b468c9fd44a542bed422" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.949ex; height:2.843ex;" alt="{\displaystyle P(n)}"></span> entsprechen. </p> <div class="mw-heading mw-heading4"><h4 id="Zusammenhang_mit_den_Pentagonalzahlen">Zusammenhang mit den Pentagonalzahlen</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Partitionsfunktion&veaction=edit&section=7" title="Abschnitt bearbeiten: Zusammenhang mit den Pentagonalzahlen" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Partitionsfunktion&action=edit&section=7" title="Quellcode des Abschnitts bearbeiten: Zusammenhang mit den Pentagonalzahlen"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Die Koeffizienten <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c(n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c(n)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/222cba36cd898cdaa8a752e7e7cb041ee5b27a3c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.211ex; height:2.843ex;" alt="{\displaystyle c(n)}"></span> von Eulers Funktion </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \prod _{k=1}^{\infty }(1-x^{k})\;=\sum _{m=-\infty }^{\infty }(-1)^{m}\cdot x^{\frac {m(3m-1)}{2}}=\sum _{n=0}^{\infty }c(n)\cdot x^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∏<!-- ∏ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞<!-- ∞ --></mi> </mrow> </munderover> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−<!-- − --></mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mo stretchy="false">)</mo> <mspace width="thickmathspace" /> <mo>=</mo> <munderover> <mo>∑<!-- ∑ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>=</mo> <mo>−<!-- − --></mo> <mi mathvariant="normal">∞<!-- ∞ --></mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞<!-- ∞ --></mi> </mrow> </munderover> <mo stretchy="false">(</mo> <mo>−<!-- − --></mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> <mo>⋅<!-- ⋅ --></mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>m</mi> <mo stretchy="false">(</mo> <mn>3</mn> <mi>m</mi> <mo>−<!-- − --></mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <mn>2</mn> </mfrac> </mrow> </msup> <mo>=</mo> <munderover> <mo>∑<!-- ∑ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞<!-- ∞ --></mi> </mrow> </munderover> <mi>c</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>⋅<!-- ⋅ --></mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \prod _{k=1}^{\infty }(1-x^{k})\;=\sum _{m=-\infty }^{\infty }(-1)^{m}\cdot x^{\frac {m(3m-1)}{2}}=\sum _{n=0}^{\infty }c(n)\cdot x^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b1060d5762be47c7d9aa0fa4e606f16c7f811584" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:52.151ex; height:6.843ex;" alt="{\displaystyle \prod _{k=1}^{\infty }(1-x^{k})\;=\sum _{m=-\infty }^{\infty }(-1)^{m}\cdot x^{\frac {m(3m-1)}{2}}=\sum _{n=0}^{\infty }c(n)\cdot x^{n}}"></span></dd></dl> <p>lassen sich mit dem <a href="/wiki/Pentagonalzahlensatz" title="Pentagonalzahlensatz">Pentagonalzahlensatz</a> von <a href="/wiki/Leonhard_Euler" title="Leonhard Euler">Leonhard Euler</a> einfach explizit berechnen. Die Folge <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c(n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c(n)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/222cba36cd898cdaa8a752e7e7cb041ee5b27a3c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.211ex; height:2.843ex;" alt="{\displaystyle c(n)}"></span> ist Folge <a href="//oeis.org/A010815" class="extiw" title="oeis:A010815">A010815</a> in <a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a> und es gilt stets <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c(n)\in \{-1,0,1\}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>∈<!-- ∈ --></mo> <mo fence="false" stretchy="false">{</mo> <mo>−<!-- − --></mo> <mn>1</mn> <mo>,</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo fence="false" stretchy="false">}</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c(n)\in \{-1,0,1\}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7667274ee6f47a36ed546d9b4ab53a947851272b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.387ex; height:2.843ex;" alt="{\displaystyle c(n)\in \{-1,0,1\}.}"></span> </p><p>Aus der Tatsache, dass Eulers Funktion multiplikativ invers zur erzeugenden Funktion der Partitionsfunktion ist, folgt, dass für die <a href="/wiki/Faltung_(Mathematik)#Diskrete_Faltung" title="Faltung (Mathematik)">diskrete Faltung</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c\ast P}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> <mo>∗<!-- ∗ --></mo> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c\ast P}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2c1d479a1785af13b10f33f656bf9e94d7a48e56" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.947ex; height:2.176ex;" alt="{\displaystyle c\ast P}"></span> und <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\in \mathbb {N} _{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>∈<!-- ∈ --></mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\in \mathbb {N} _{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0cf5c8ad993619a325ca57a25c22cdc75a460f88" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.968ex; height:2.509ex;" alt="{\displaystyle n\in \mathbb {N} _{0}}"></span> gilt </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (c\ast P)(n)=\sum _{r\in \mathbb {Z} }c(r)\cdot P(n-r)=\sum _{r=0}^{n}P(r)\cdot c(n-r)={\begin{cases}1\;(n=0)\\0\;(n\neq 0).\end{cases}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>c</mi> <mo>∗<!-- ∗ --></mo> <mi>P</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munder> <mo>∑<!-- ∑ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> <mo>∈<!-- ∈ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mrow> </munder> <mi>c</mi> <mo stretchy="false">(</mo> <mi>r</mi> <mo stretchy="false">)</mo> <mo>⋅<!-- ⋅ --></mo> <mi>P</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−<!-- − --></mo> <mi>r</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munderover> <mo>∑<!-- ∑ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mi>P</mi> <mo stretchy="false">(</mo> <mi>r</mi> <mo stretchy="false">)</mo> <mo>⋅<!-- ⋅ --></mo> <mi>c</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−<!-- − --></mo> <mi>r</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <mn>1</mn> <mspace width="thickmathspace" /> <mo stretchy="false">(</mo> <mi>n</mi> <mo>=</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> <mspace width="thickmathspace" /> <mo stretchy="false">(</mo> <mi>n</mi> <mo>≠<!-- ≠ --></mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>.</mo> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (c\ast P)(n)=\sum _{r\in \mathbb {Z} }c(r)\cdot P(n-r)=\sum _{r=0}^{n}P(r)\cdot c(n-r)={\begin{cases}1\;(n=0)\\0\;(n\neq 0).\end{cases}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/989624bdcb38b17dbb90dcd5c080c4557dac572a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:67.919ex; height:7.009ex;" alt="{\displaystyle (c\ast P)(n)=\sum _{r\in \mathbb {Z} }c(r)\cdot P(n-r)=\sum _{r=0}^{n}P(r)\cdot c(n-r)={\begin{cases}1\;(n=0)\\0\;(n\neq 0).\end{cases}}}"></span></dd></dl> <p>Die Summation muss nur über <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r\in \{0,1,\ldots n\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo>∈<!-- ∈ --></mo> <mo fence="false" stretchy="false">{</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mo>…<!-- … --></mo> <mi>n</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r\in \{0,1,\ldots n\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3d54bcf72678b1aef680214dbe8e9a7220443903" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.112ex; height:2.843ex;" alt="{\displaystyle r\in \{0,1,\ldots n\}}"></span> erstreckt werden, da beide Folgen als Koeffizientenfolgen ihrer jeweiligen Funktion an negativen Stellen gleich Null sind. </p> <div class="mw-heading mw-heading4"><h4 id="Rekursionsformel_aus_dem_Pentagonalzahlensatz">Rekursionsformel aus dem Pentagonalzahlensatz</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Partitionsfunktion&veaction=edit&section=8" title="Abschnitt bearbeiten: Rekursionsformel aus dem Pentagonalzahlensatz" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Partitionsfunktion&action=edit&section=8" title="Quellcode des Abschnitts bearbeiten: Rekursionsformel aus dem Pentagonalzahlensatz"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Aus der im vorigen Unterabschnitt angegebenen Faltungsbeziehung zu den Koeffizienten <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c(n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c(n)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/222cba36cd898cdaa8a752e7e7cb041ee5b27a3c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.211ex; height:2.843ex;" alt="{\displaystyle c(n)}"></span> folgt für <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\in \mathbb {N} \setminus \{0\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>∈<!-- ∈ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> <mo class="MJX-variant">∖<!-- ∖ --></mo> <mo fence="false" stretchy="false">{</mo> <mn>0</mn> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\in \mathbb {N} \setminus \{0\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a0055197414c7cf4bfa79fd73e0c943d14251877" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.596ex; height:2.843ex;" alt="{\displaystyle n\in \mathbb {N} \setminus \{0\}}"></span> die Rekursionsformel </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(n)=-\sum _{r=1}^{n}c(r)\cdot P(n-r)=P(n-1)+P(n-2)-P(n-5)-P(n-7)+\cdots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo>−<!-- − --></mo> <munderover> <mo>∑<!-- ∑ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mi>c</mi> <mo stretchy="false">(</mo> <mi>r</mi> <mo stretchy="false">)</mo> <mo>⋅<!-- ⋅ --></mo> <mi>P</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−<!-- − --></mo> <mi>r</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>P</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−<!-- − --></mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>+</mo> <mi>P</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−<!-- − --></mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo>−<!-- − --></mo> <mi>P</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−<!-- − --></mo> <mn>5</mn> <mo stretchy="false">)</mo> <mo>−<!-- − --></mo> <mi>P</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−<!-- − --></mo> <mn>7</mn> <mo stretchy="false">)</mo> <mo>+</mo> <mo>⋯<!-- ⋯ --></mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(n)=-\sum _{r=1}^{n}c(r)\cdot P(n-r)=P(n-1)+P(n-2)-P(n-5)-P(n-7)+\cdots }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5aeff93b87c80ce4f40bd6ed550ec7089aba4cd6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:81.36ex; height:6.843ex;" alt="{\displaystyle P(n)=-\sum _{r=1}^{n}c(r)\cdot P(n-r)=P(n-1)+P(n-2)-P(n-5)-P(n-7)+\cdots }"></span></dd></dl> <p>für die Partitionsfunktion. </p> <div class="mw-heading mw-heading3"><h3 id="Berechnung_mit_analytischer_Zahlentheorie">Berechnung mit analytischer Zahlentheorie</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Partitionsfunktion&veaction=edit&section=9" title="Abschnitt bearbeiten: Berechnung mit analytischer Zahlentheorie" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Partitionsfunktion&action=edit&section=9" title="Quellcode des Abschnitts bearbeiten: Berechnung mit analytischer Zahlentheorie"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Eine Möglichkeit zur direkten Berechnung liefert die aus der erzeugenden Funktion hergeleitete Formel </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(n)={\frac {1}{\pi {\sqrt {2}}}}\sum _{k=1}^{\infty }A_{k}(n)\;{\sqrt {k}}\;{\frac {\mathrm {d} }{\mathrm {d} n}}\left\{{\frac {\sinh {\bigl [}{\frac {\pi }{k}}{\sqrt {{\frac {2}{3}}{\bigl (}n-{\frac {1}{24}}{\bigr )}}}{\bigr ]}}{\sqrt {n-{\frac {1}{24}}}}}\right\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>π<!-- π --></mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> </mrow> </mfrac> </mrow> <munderover> <mo>∑<!-- ∑ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞<!-- ∞ --></mi> </mrow> </munderover> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mspace width="thickmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>k</mi> </msqrt> </mrow> <mspace width="thickmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>n</mi> </mrow> </mfrac> </mrow> <mrow> <mo>{</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>sinh</mi> <mo>⁡<!-- --></mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">[</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π<!-- π --></mi> <mi>k</mi> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <mn>3</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mi>n</mi> <mo>−<!-- − --></mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>24</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> </msqrt> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">]</mo> </mrow> </mrow> </mrow> <msqrt> <mi>n</mi> <mo>−<!-- − --></mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>24</mn> </mfrac> </mrow> </msqrt> </mfrac> </mrow> <mo>}</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(n)={\frac {1}{\pi {\sqrt {2}}}}\sum _{k=1}^{\infty }A_{k}(n)\;{\sqrt {k}}\;{\frac {\mathrm {d} }{\mathrm {d} n}}\left\{{\frac {\sinh {\bigl [}{\frac {\pi }{k}}{\sqrt {{\frac {2}{3}}{\bigl (}n-{\frac {1}{24}}{\bigr )}}}{\bigr ]}}{\sqrt {n-{\frac {1}{24}}}}}\right\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/45472ef01140b4a9784534f0ad4be938f0a5fc15" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.671ex; width:57.875ex; height:10.509ex;" alt="{\displaystyle P(n)={\frac {1}{\pi {\sqrt {2}}}}\sum _{k=1}^{\infty }A_{k}(n)\;{\sqrt {k}}\;{\frac {\mathrm {d} }{\mathrm {d} n}}\left\{{\frac {\sinh {\bigl [}{\frac {\pi }{k}}{\sqrt {{\frac {2}{3}}{\bigl (}n-{\frac {1}{24}}{\bigr )}}}{\bigr ]}}{\sqrt {n-{\frac {1}{24}}}}}\right\}}"></span></dd></dl> <p>mit </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{k}(n)=\!\!\!\!\sum _{0\leq m<k \atop \operatorname {ggT} (m,k)=1}\!\!\!\!\exp \left\{{\frac {\pi i}{k}}\left[s(m,k)-2nm\right]\right\}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mspace width="negativethinmathspace" /> <mspace width="negativethinmathspace" /> <mspace width="negativethinmathspace" /> <mspace width="negativethinmathspace" /> <munder> <mo>∑<!-- ∑ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac linethickness="0"> <mrow> <mn>0</mn> <mo>≤<!-- ≤ --></mo> <mi>m</mi> <mo><</mo> <mi>k</mi> </mrow> <mrow> <mi>ggT</mi> <mo>⁡<!-- --></mo> <mo stretchy="false">(</mo> <mi>m</mi> <mo>,</mo> <mi>k</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1</mn> </mrow> </mfrac> </mrow> </munder> <mspace width="negativethinmathspace" /> <mspace width="negativethinmathspace" /> <mspace width="negativethinmathspace" /> <mspace width="negativethinmathspace" /> <mi>exp</mi> <mo>⁡<!-- --></mo> <mrow> <mo>{</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>π<!-- π --></mi> <mi>i</mi> </mrow> <mi>k</mi> </mfrac> </mrow> <mrow> <mo>[</mo> <mrow> <mi>s</mi> <mo stretchy="false">(</mo> <mi>m</mi> <mo>,</mo> <mi>k</mi> <mo stretchy="false">)</mo> <mo>−<!-- − --></mo> <mn>2</mn> <mi>n</mi> <mi>m</mi> </mrow> <mo>]</mo> </mrow> </mrow> <mo>}</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A_{k}(n)=\!\!\!\!\sum _{0\leq m<k \atop \operatorname {ggT} (m,k)=1}\!\!\!\!\exp \left\{{\frac {\pi i}{k}}\left[s(m,k)-2nm\right]\right\}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a01001ab87060fd9f6030db66116f4ea4f179805" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.838ex; width:41.222ex; height:8.509ex;" alt="{\displaystyle A_{k}(n)=\!\!\!\!\sum _{0\leq m<k \atop \operatorname {ggT} (m,k)=1}\!\!\!\!\exp \left\{{\frac {\pi i}{k}}\left[s(m,k)-2nm\right]\right\}.}"></span></dd></dl> <p>und </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s(m,k)=\sum _{j=1}^{k-1}j\left({\frac {mj}{k}}-\left\lfloor {\frac {mj}{k}}\right\rfloor -{\frac {1}{2}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> <mo stretchy="false">(</mo> <mi>m</mi> <mo>,</mo> <mi>k</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munderover> <mo>∑<!-- ∑ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>−<!-- − --></mo> <mn>1</mn> </mrow> </munderover> <mi>j</mi> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>m</mi> <mi>j</mi> </mrow> <mi>k</mi> </mfrac> </mrow> <mo>−<!-- − --></mo> <mrow> <mo>⌊</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>m</mi> <mi>j</mi> </mrow> <mi>k</mi> </mfrac> </mrow> <mo>⌋</mo> </mrow> <mo>−<!-- − --></mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s(m,k)=\sum _{j=1}^{k-1}j\left({\frac {mj}{k}}-\left\lfloor {\frac {mj}{k}}\right\rfloor -{\frac {1}{2}}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/04c10c01d090cb345cff678a939ebc9022696089" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:36.851ex; height:7.676ex;" alt="{\displaystyle s(m,k)=\sum _{j=1}^{k-1}j\left({\frac {mj}{k}}-\left\lfloor {\frac {mj}{k}}\right\rfloor -{\frac {1}{2}}\right)}"></span></dd></dl> <p>die <a href="/wiki/Hans_Rademacher" title="Hans Rademacher">Hans Rademacher</a>, aufbauend auf Erkenntnissen von <a href="/wiki/S._Ramanujan" class="mw-redirect" title="S. Ramanujan">S. Ramanujan</a> und <a href="/wiki/Godfrey_Harold_Hardy" title="Godfrey Harold Hardy">Godfrey Harold Hardy</a>, fand. </p> <div class="mw-heading mw-heading3"><h3 id="Berechnung_mit_algebraischer_Zahlentheorie">Berechnung mit algebraischer Zahlentheorie</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Partitionsfunktion&veaction=edit&section=10" title="Abschnitt bearbeiten: Berechnung mit algebraischer Zahlentheorie" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Partitionsfunktion&action=edit&section=10" title="Quellcode des Abschnitts bearbeiten: Berechnung mit algebraischer Zahlentheorie"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Eine algebraische, geschlossene Form von <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(n)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5e303d2c14cd399b6f52b468c9fd44a542bed422" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.949ex; height:2.843ex;" alt="{\displaystyle P(n)}"></span>, die ohne <a href="/wiki/Unendliche_Reihe" class="mw-redirect" title="Unendliche Reihe">unendliche Reihenentwicklung</a> auskommt, wurde 2011 von <a href="/wiki/Jan_Hendrik_Bruinier" title="Jan Hendrik Bruinier">Jan Hendrik Bruinier</a> und <a href="/wiki/Ken_Ono" title="Ken Ono">Ken Ono</a> veröffentlicht.<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> Genauer gesagt geben Bruinier und Ono eine Funktion <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8752c7023b4b3286800fe3238271bbca681219ed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.838ex; height:2.509ex;" alt="{\displaystyle Q}"></span> an, so dass sich für jede natürliche Zahl <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span> eine endliche Anzahl <a href="/wiki/Algebraische_Zahl" title="Algebraische Zahl">algebraischer Zahlen</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha _{1},\alpha _{2},\ldots ,\alpha _{m}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>α<!-- α --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>α<!-- α --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>…<!-- … --></mo> <mo>,</mo> <msub> <mi>α<!-- α --></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha _{1},\alpha _{2},\ldots ,\alpha _{m}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/29f50c85e90eabc65a77ffd188ea077ff455c41d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.459ex; height:2.009ex;" alt="{\displaystyle \alpha _{1},\alpha _{2},\ldots ,\alpha _{m}}"></span> mit </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(n)={\frac {1}{24n-1}}\left(Q(\alpha _{1})+Q(\alpha _{2})+\ldots +Q(\alpha _{m})\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>24</mn> <mi>n</mi> <mo>−<!-- − --></mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <mi>Q</mi> <mo stretchy="false">(</mo> <msub> <mi>α<!-- α --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>+</mo> <mi>Q</mi> <mo stretchy="false">(</mo> <msub> <mi>α<!-- α --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>+</mo> <mo>…<!-- … --></mo> <mo>+</mo> <mi>Q</mi> <mo stretchy="false">(</mo> <msub> <mi>α<!-- α --></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(n)={\frac {1}{24n-1}}\left(Q(\alpha _{1})+Q(\alpha _{2})+\ldots +Q(\alpha _{m})\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c62aaa4720b1f6af4b7047e97583253723837465" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:49.237ex; height:5.343ex;" alt="{\displaystyle P(n)={\frac {1}{24n-1}}\left(Q(\alpha _{1})+Q(\alpha _{2})+\ldots +Q(\alpha _{m})\right)}"></span></dd></dl> <p>finden lassen. Darüber hinaus gilt, dass auch alle Werte <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q(\alpha _{i})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Q</mi> <mo stretchy="false">(</mo> <msub> <mi>α<!-- α --></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q(\alpha _{i})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ade17b8c093633db37830abfa833cbe18cb5be59" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.935ex; height:2.843ex;" alt="{\displaystyle Q(\alpha _{i})}"></span> algebraisch sind. </p><p>Dieses theoretische Ergebnis führt nur in Spezialfällen (z. B. über daraus ableitbare Kongruenzen) zu einer schnelleren Berechnung der Partitionsfunktion. </p> <div class="mw-heading mw-heading3"><h3 id="Kongruenzen">Kongruenzen</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Partitionsfunktion&veaction=edit&section=11" title="Abschnitt bearbeiten: Kongruenzen" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Partitionsfunktion&action=edit&section=11" title="Quellcode des Abschnitts bearbeiten: Kongruenzen"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <table class="wikitable float-right"> <caption> </caption> <tbody><tr> <th><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span></th> <th><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(n)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5e303d2c14cd399b6f52b468c9fd44a542bed422" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.949ex; height:2.843ex;" alt="{\displaystyle P(n)}"></span></th> <th>Kongruenzen </th></tr> <tr> <td>1</td> <td>1</td> <td> </td></tr> <tr> <td>2</td> <td>2</td> <td> </td></tr> <tr> <td>3</td> <td>3</td> <td> </td></tr> <tr> <td>4</td> <td>5</td> <td>mod 5 </td></tr> <tr> <td>5</td> <td>7</td> <td>mod 7 </td></tr> <tr> <td>6</td> <td>11</td> <td>mod 11 </td></tr> <tr> <td>7</td> <td>15</td> <td> </td></tr> <tr> <td>8</td> <td>22</td> <td> </td></tr> <tr> <td>9</td> <td>30</td> <td>mod 5 </td></tr> <tr> <td>10</td> <td>42</td> <td> </td></tr> <tr> <td>11</td> <td>56</td> <td> </td></tr> <tr> <td>12</td> <td>77</td> <td>mod 7 </td></tr> <tr> <td>13</td> <td>101</td> <td> </td></tr> <tr> <td>14</td> <td>135</td> <td>mod 5 </td></tr> <tr> <td>15</td> <td>176</td> <td> </td></tr> <tr> <td>16</td> <td>231</td> <td> </td></tr> <tr> <td>17</td> <td>297</td> <td>mod 11 </td></tr> <tr> <td>18</td> <td>385</td> <td> </td></tr> <tr> <td>19</td> <td>490</td> <td>mod 5 und 7 </td></tr> <tr> <td>20</td> <td>627</td> <td> </td></tr></tbody></table> <p>Ramanujan entdeckte bei seinen Studien eine Gesetzmäßigkeit. Beginnt man mit der 4 und springt um 5, so erhält man immer Vielfache der Sprungzahl 5 als Zerlegungszahlen. Beginnt man bei der 6 und springt um 11, so erhält man Vielfache von 11. Ramanujan entdeckte weitere derartige Beziehungen, auch Kongruenzen genannt, als er die Potenzen der Primzahlen 5, 7 und 11 sowie deren Produkte als Sprungzahlen untersuchte. Der amerikanische Zahlentheoretiker Ken Ono konnte zeigen, dass es für alle Primzahlen größer 3 Kongruenzen gibt. Ob dies für die beiden kleinsten Primzahlen, die 2 und 3, und deren Vielfache ebenso gilt, konnte Ono nicht nachweisen. Folgende Kongruenzen gehen auf Ramanujan zurück: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}P(5k+4)&\equiv 0\mod 5\\P(7k+5)&\equiv 0\mod 7\\P(11k+6)&\equiv 0\mod {11}\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>P</mi> <mo stretchy="false">(</mo> <mn>5</mn> <mi>k</mi> <mo>+</mo> <mn>4</mn> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>≡<!-- ≡ --></mo> <mn>0</mn> <mspace width="1em" /> <mi>mod</mi> <mspace width="thinmathspace" /> <mspace width="thinmathspace" /> <mn>5</mn> </mtd> </mtr> <mtr> <mtd> <mi>P</mi> <mo stretchy="false">(</mo> <mn>7</mn> <mi>k</mi> <mo>+</mo> <mn>5</mn> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>≡<!-- ≡ --></mo> <mn>0</mn> <mspace width="1em" /> <mi>mod</mi> <mspace width="thinmathspace" /> <mspace width="thinmathspace" /> <mn>7</mn> </mtd> </mtr> <mtr> <mtd> <mi>P</mi> <mo stretchy="false">(</mo> <mn>11</mn> <mi>k</mi> <mo>+</mo> <mn>6</mn> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>≡<!-- ≡ --></mo> <mn>0</mn> <mspace width="1em" /> <mi>mod</mi> <mspace width="thinmathspace" /> <mspace width="thinmathspace" /> <mn>11</mn> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}P(5k+4)&\equiv 0\mod 5\\P(7k+5)&\equiv 0\mod 7\\P(11k+6)&\equiv 0\mod {11}\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3d08b19279636a2a2208561a9dfba1db73e656b4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:26.693ex; height:9.176ex;" alt="{\displaystyle {\begin{aligned}P(5k+4)&\equiv 0\mod 5\\P(7k+5)&\equiv 0\mod 7\\P(11k+6)&\equiv 0\mod {11}\end{aligned}}}"></span></dd></dl> <p><a href="/wiki/A._O._L._Atkin" title="A. O. L. Atkin">A. O. L. Atkin</a> fand folgende Kongruenz: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(17303k+237)\equiv 0\mod {13}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <mn>17303</mn> <mi>k</mi> <mo>+</mo> <mn>237</mn> <mo stretchy="false">)</mo> <mo>≡<!-- ≡ --></mo> <mn>0</mn> <mspace width="1em" /> <mi>mod</mi> <mspace width="thinmathspace" /> <mspace width="thinmathspace" /> <mn>13</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(17303k+237)\equiv 0\mod {13}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3bc774309bd6cf3497e96c22faef258157cf498e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:31.754ex; height:2.843ex;" alt="{\displaystyle P(17303k+237)\equiv 0\mod {13}}"></span></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Ferrers-Diagramme">Ferrers-Diagramme</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Partitionsfunktion&veaction=edit&section=12" title="Abschnitt bearbeiten: Ferrers-Diagramme" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Partitionsfunktion&action=edit&section=12" title="Quellcode des Abschnitts bearbeiten: Ferrers-Diagramme"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>→ Im Artikel <a href="/wiki/Young-Tableau" title="Young-Tableau">Young-Tableau</a> wird ein ähnlicher Diagrammtyp ausführlich beschrieben, der wie die hier beschriebenen Ferrers-Diagramme eine Partition eindeutig bestimmt und vor allem in der <a href="/wiki/Darstellungstheorie" title="Darstellungstheorie">Darstellungstheorie</a> verwendet wird. </p><p>Die Zahlpartition <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 6+4+3+1=14}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>6</mn> <mo>+</mo> <mn>4</mn> <mo>+</mo> <mn>3</mn> <mo>+</mo> <mn>1</mn> <mo>=</mo> <mn>14</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 6+4+3+1=14}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68b560bba6f33ef0f050410daa907a736c6f88ae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:18.594ex; height:2.343ex;" alt="{\displaystyle 6+4+3+1=14}"></span> kann durch folgendes Diagramm, das als <i>Ferrers-Diagramm</i> bezeichnet wird, dargestellt werden. Diese Diagramme wurden zu Ehren von Norman Macleod Ferrers benannt.<sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> </p> <table> <tbody><tr style="vertical-align:top; text-align:left;"> <td><span typeof="mw:File"><a href="/wiki/Datei:GrayDot.svg" class="mw-file-description" title="*"><img alt="*" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/16px-GrayDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/24px-GrayDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/32px-GrayDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span><span typeof="mw:File"><a href="/wiki/Datei:GrayDot.svg" class="mw-file-description" title="*"><img alt="*" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/16px-GrayDot.svg.png" 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typeof="mw:File"><a href="/wiki/Datei:GrayDot.svg" class="mw-file-description" title="*"><img alt="*" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/16px-GrayDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/24px-GrayDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/32px-GrayDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span><br /><span typeof="mw:File"><a href="/wiki/Datei:GrayDot.svg" class="mw-file-description" title="*"><img alt="*" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/16px-GrayDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/24px-GrayDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/32px-GrayDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span><br /><span typeof="mw:File"><a href="/wiki/Datei:GrayDot.svg" class="mw-file-description" title="*"><img alt="*" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/16px-GrayDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/24px-GrayDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/32px-GrayDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span> </td></tr> <tr style="vertical-align:top; text-align:center;"> <td>6 + 4 + 3 + 1 </td></tr></tbody></table> <p>Die 14 Kreise werden in 4 Spalten für die 4 Summanden der Partition aufgereiht, wobei die Spalten von links nach rechts nie <i>höher</i> werden. Es wird auch häufig die umgekehrte Konvention verwendet, bei der die Säulen von Kreisen auf der Grundlinie stehen und von links nach rechts nie <i>niedriger</i> werden. Die 5 Partitionen von 4 sind nachfolgend als Ferrers-Diagramme dargestellt: </p> <table> <tbody><tr style="vertical-align:top; text-align:left;"> <td><span typeof="mw:File"><a href="/wiki/Datei:GrayDot.svg" class="mw-file-description" title="*"><img alt="*" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/16px-GrayDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/24px-GrayDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/32px-GrayDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span><br /><span typeof="mw:File"><a href="/wiki/Datei:GrayDot.svg" class="mw-file-description" title="*"><img alt="*" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/16px-GrayDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/24px-GrayDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/32px-GrayDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span><br /><span typeof="mw:File"><a href="/wiki/Datei:GrayDot.svg" class="mw-file-description" title="*"><img alt="*" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/16px-GrayDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/24px-GrayDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/32px-GrayDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span><br /><span typeof="mw:File"><a href="/wiki/Datei:GrayDot.svg" class="mw-file-description" title="*"><img alt="*" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/16px-GrayDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/24px-GrayDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/32px-GrayDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span></td> <td> </td> <td><span typeof="mw:File"><a href="/wiki/Datei:GrayDot.svg" class="mw-file-description" title="*"><img alt="*" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/16px-GrayDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/24px-GrayDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/32px-GrayDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span><span typeof="mw:File"><a href="/wiki/Datei:GrayDot.svg" class="mw-file-description" title="*"><img alt="*" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/16px-GrayDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/24px-GrayDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/32px-GrayDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span><br /><span typeof="mw:File"><a href="/wiki/Datei:GrayDot.svg" class="mw-file-description" title="*"><img alt="*" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/16px-GrayDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/24px-GrayDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/32px-GrayDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span><br /><span typeof="mw:File"><a href="/wiki/Datei:GrayDot.svg" class="mw-file-description" title="*"><img alt="*" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/16px-GrayDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/24px-GrayDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/32px-GrayDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span></td> <td> </td> <td><span typeof="mw:File"><a href="/wiki/Datei:GrayDot.svg" class="mw-file-description" title="*"><img alt="*" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/16px-GrayDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/24px-GrayDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/32px-GrayDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span><span typeof="mw:File"><a href="/wiki/Datei:GrayDot.svg" class="mw-file-description" title="*"><img alt="*" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/16px-GrayDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/24px-GrayDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/32px-GrayDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span><br /><span typeof="mw:File"><a href="/wiki/Datei:GrayDot.svg" class="mw-file-description" title="*"><img alt="*" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/16px-GrayDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/24px-GrayDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/32px-GrayDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span><span typeof="mw:File"><a href="/wiki/Datei:GrayDot.svg" class="mw-file-description" title="*"><img alt="*" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/16px-GrayDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/24px-GrayDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/32px-GrayDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span></td> <td> </td> <td><span typeof="mw:File"><a href="/wiki/Datei:GrayDot.svg" class="mw-file-description" title="*"><img alt="*" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/16px-GrayDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/24px-GrayDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/32px-GrayDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span><span typeof="mw:File"><a href="/wiki/Datei:GrayDot.svg" class="mw-file-description" title="*"><img alt="*" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/16px-GrayDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/24px-GrayDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/32px-GrayDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span><span typeof="mw:File"><a href="/wiki/Datei:GrayDot.svg" class="mw-file-description" title="*"><img alt="*" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/16px-GrayDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/24px-GrayDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/32px-GrayDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span><br /><span typeof="mw:File"><a href="/wiki/Datei:GrayDot.svg" class="mw-file-description" title="*"><img alt="*" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/16px-GrayDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/24px-GrayDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/32px-GrayDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span></td> <td> </td> <td><span typeof="mw:File"><a href="/wiki/Datei:GrayDot.svg" class="mw-file-description" title="*"><img alt="*" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/16px-GrayDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/24px-GrayDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/32px-GrayDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span><span typeof="mw:File"><a href="/wiki/Datei:GrayDot.svg" class="mw-file-description" title="*"><img alt="*" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/16px-GrayDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/24px-GrayDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/32px-GrayDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span><span typeof="mw:File"><a href="/wiki/Datei:GrayDot.svg" class="mw-file-description" title="*"><img alt="*" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/16px-GrayDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/24px-GrayDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/32px-GrayDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span><span typeof="mw:File"><a href="/wiki/Datei:GrayDot.svg" class="mw-file-description" title="*"><img alt="*" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/16px-GrayDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/24px-GrayDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/32px-GrayDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span> </td></tr> <tr style="vertical-align:top; text-align:center;"> <td>4</td> <td>= </td> <td>3 + 1</td> <td>= </td> <td>2 + 2</td> <td>= </td> <td>2 + 1 + 1</td> <td>= </td> <td>1 + 1 + 1 + 1 </td></tr></tbody></table> <div class="mw-heading mw-heading3"><h3 id="Konjugierte_Partition">Konjugierte Partition</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Partitionsfunktion&veaction=edit&section=13" title="Abschnitt bearbeiten: Konjugierte Partition" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Partitionsfunktion&action=edit&section=13" title="Quellcode des Abschnitts bearbeiten: Konjugierte Partition"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Wenn wir das Diagramm der Partition <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 6+4+3+1=14}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>6</mn> <mo>+</mo> <mn>4</mn> <mo>+</mo> <mn>3</mn> <mo>+</mo> <mn>1</mn> <mo>=</mo> <mn>14</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 6+4+3+1=14}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68b560bba6f33ef0f050410daa907a736c6f88ae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:18.594ex; height:2.343ex;" alt="{\displaystyle 6+4+3+1=14}"></span> an seiner Hauptdiagonale spiegeln, erhalten wir eine andere Partition von 14: </p> <table> <tbody><tr style="vertical-align:top; text-align:left;"> <td><span typeof="mw:File"><a href="/wiki/Datei:RedDot.svg" class="mw-file-description" title="*"><img alt="*" src="//upload.wikimedia.org/wikipedia/commons/thumb/e/ec/RedDot.svg/16px-RedDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/ec/RedDot.svg/24px-RedDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/ec/RedDot.svg/32px-RedDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span><span typeof="mw:File"><a href="/wiki/Datei:GrayDot.svg" class="mw-file-description" title="*"><img alt="*" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/16px-GrayDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/24px-GrayDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/32px-GrayDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span><span typeof="mw:File"><a href="/wiki/Datei:GrayDot.svg" class="mw-file-description" title="*"><img alt="*" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/16px-GrayDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/24px-GrayDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/32px-GrayDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span><span typeof="mw:File"><a href="/wiki/Datei:GrayDot.svg" class="mw-file-description" title="*"><img alt="*" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/16px-GrayDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/24px-GrayDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/32px-GrayDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span><br /><span typeof="mw:File"><a href="/wiki/Datei:GrayDot.svg" class="mw-file-description" title="*"><img alt="*" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/16px-GrayDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/24px-GrayDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/32px-GrayDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span><span typeof="mw:File"><a href="/wiki/Datei:RedDot.svg" class="mw-file-description" title="*"><img alt="*" src="//upload.wikimedia.org/wikipedia/commons/thumb/e/ec/RedDot.svg/16px-RedDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/ec/RedDot.svg/24px-RedDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/ec/RedDot.svg/32px-RedDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span><span typeof="mw:File"><a href="/wiki/Datei:GrayDot.svg" class="mw-file-description" title="*"><img alt="*" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/16px-GrayDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/24px-GrayDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/32px-GrayDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span><br /><span typeof="mw:File"><a href="/wiki/Datei:GrayDot.svg" class="mw-file-description" title="*"><img alt="*" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/16px-GrayDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/24px-GrayDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/32px-GrayDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span><span typeof="mw:File"><a href="/wiki/Datei:GrayDot.svg" class="mw-file-description" title="*"><img alt="*" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/16px-GrayDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/24px-GrayDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/32px-GrayDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span><span typeof="mw:File"><a href="/wiki/Datei:RedDot.svg" class="mw-file-description" title="*"><img alt="*" src="//upload.wikimedia.org/wikipedia/commons/thumb/e/ec/RedDot.svg/16px-RedDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/ec/RedDot.svg/24px-RedDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/ec/RedDot.svg/32px-RedDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span><br /><span typeof="mw:File"><a href="/wiki/Datei:GrayDot.svg" class="mw-file-description" title="*"><img alt="*" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/16px-GrayDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/24px-GrayDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/32px-GrayDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span><span typeof="mw:File"><a href="/wiki/Datei:GrayDot.svg" class="mw-file-description" title="*"><img alt="*" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/16px-GrayDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/24px-GrayDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/32px-GrayDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span><br /><span typeof="mw:File"><a href="/wiki/Datei:GrayDot.svg" class="mw-file-description" title="*"><img alt="*" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/16px-GrayDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/24px-GrayDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/32px-GrayDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span><br /><span typeof="mw:File"><a href="/wiki/Datei:GrayDot.svg" class="mw-file-description" title="*"><img alt="*" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/16px-GrayDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/24px-GrayDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/32px-GrayDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span> </td> <td style="vertical-align:middle;">↔ </td> <td><span typeof="mw:File"><a href="/wiki/Datei:RedDot.svg" class="mw-file-description" title="*"><img alt="*" src="//upload.wikimedia.org/wikipedia/commons/thumb/e/ec/RedDot.svg/16px-RedDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/ec/RedDot.svg/24px-RedDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/ec/RedDot.svg/32px-RedDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span><span typeof="mw:File"><a href="/wiki/Datei:GrayDot.svg" class="mw-file-description" title="*"><img alt="*" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/16px-GrayDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/24px-GrayDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/32px-GrayDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span><span typeof="mw:File"><a href="/wiki/Datei:GrayDot.svg" class="mw-file-description" title="*"><img alt="*" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/16px-GrayDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/24px-GrayDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/32px-GrayDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span><span typeof="mw:File"><a href="/wiki/Datei:GrayDot.svg" class="mw-file-description" title="*"><img alt="*" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/16px-GrayDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/24px-GrayDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/32px-GrayDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span><span typeof="mw:File"><a href="/wiki/Datei:GrayDot.svg" class="mw-file-description" title="*"><img alt="*" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/16px-GrayDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/24px-GrayDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/32px-GrayDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span><span typeof="mw:File"><a href="/wiki/Datei:GrayDot.svg" class="mw-file-description" title="*"><img alt="*" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/16px-GrayDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/24px-GrayDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/32px-GrayDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span><br /><span typeof="mw:File"><a href="/wiki/Datei:GrayDot.svg" class="mw-file-description" title="*"><img alt="*" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/16px-GrayDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/24px-GrayDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/32px-GrayDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span><span typeof="mw:File"><a href="/wiki/Datei:RedDot.svg" class="mw-file-description" title="*"><img alt="*" src="//upload.wikimedia.org/wikipedia/commons/thumb/e/ec/RedDot.svg/16px-RedDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/ec/RedDot.svg/24px-RedDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/ec/RedDot.svg/32px-RedDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span><span typeof="mw:File"><a href="/wiki/Datei:GrayDot.svg" class="mw-file-description" title="*"><img alt="*" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/16px-GrayDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/24px-GrayDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/32px-GrayDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span><span typeof="mw:File"><a href="/wiki/Datei:GrayDot.svg" class="mw-file-description" title="*"><img alt="*" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/16px-GrayDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/24px-GrayDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/32px-GrayDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span><br /><span typeof="mw:File"><a href="/wiki/Datei:GrayDot.svg" class="mw-file-description" title="*"><img alt="*" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/16px-GrayDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/24px-GrayDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/32px-GrayDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span><span typeof="mw:File"><a href="/wiki/Datei:GrayDot.svg" class="mw-file-description" title="*"><img alt="*" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/16px-GrayDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/24px-GrayDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/32px-GrayDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span><span typeof="mw:File"><a href="/wiki/Datei:RedDot.svg" class="mw-file-description" title="*"><img alt="*" src="//upload.wikimedia.org/wikipedia/commons/thumb/e/ec/RedDot.svg/16px-RedDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/ec/RedDot.svg/24px-RedDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/ec/RedDot.svg/32px-RedDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span><br /><span typeof="mw:File"><a href="/wiki/Datei:GrayDot.svg" class="mw-file-description" title="*"><img alt="*" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/16px-GrayDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/24px-GrayDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/32px-GrayDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span> </td></tr> <tr style="vertical-align:top; text-align:center;"> <td>6 + 4 + 3 + 1 </td> <td>= </td> <td>4 + 3 + 3 + 2 + 1 + 1 </td></tr></tbody></table> <p>Indem wir so Reihen in Spalten verwandeln, erhalten wir die Partition <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 4+3+3+2+1+1=14}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>4</mn> <mo>+</mo> <mn>3</mn> <mo>+</mo> <mn>3</mn> <mo>+</mo> <mn>2</mn> <mo>+</mo> <mn>1</mn> <mo>+</mo> <mn>1</mn> <mo>=</mo> <mn>14</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 4+3+3+2+1+1=14}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4961cdc3485ab1e0dc126df742ca54d5db8b5266" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:26.6ex; height:2.343ex;" alt="{\displaystyle 4+3+3+2+1+1=14}"></span>. Sie heißt die zu <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 6+4+3+1=14}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>6</mn> <mo>+</mo> <mn>4</mn> <mo>+</mo> <mn>3</mn> <mo>+</mo> <mn>1</mn> <mo>=</mo> <mn>14</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 6+4+3+1=14}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68b560bba6f33ef0f050410daa907a736c6f88ae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:18.594ex; height:2.343ex;" alt="{\displaystyle 6+4+3+1=14}"></span> <i>konjugierte Partition</i>.<sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> Unter den Partitionen von 4 sind <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1+1+1+1=4}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>+</mo> <mn>1</mn> <mo>+</mo> <mn>1</mn> <mo>+</mo> <mn>1</mn> <mo>=</mo> <mn>4</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1+1+1+1=4}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6c4b62d411406c8e62356e4e47f9707851ed33c4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:17.432ex; height:2.343ex;" alt="{\displaystyle 1+1+1+1=4}"></span> und <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 4=4}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>4</mn> <mo>=</mo> <mn>4</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 4=4}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cc2a05e2a43d0c423a81f3c962da43c1cac37c3a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.423ex; height:2.176ex;" alt="{\displaystyle 4=4}"></span>; <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 3+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>3</mn> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 3+1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4ecc518bc01cc01d7394034cdc5db9c3eb3e20de" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.165ex; height:2.343ex;" alt="{\displaystyle 3+1}"></span> und <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2+1+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mo>+</mo> <mn>1</mn> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2+1+1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/131e69d778f8d51cc9d1993ef6e02eeadb8cee71" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.168ex; height:2.343ex;" alt="{\displaystyle 2+1+1}"></span> jeweils konjugiert zueinander. Besonders interessant sind Partitionen wie <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 3+2+1=6}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>3</mn> <mo>+</mo> <mn>2</mn> <mo>+</mo> <mn>1</mn> <mo>=</mo> <mn>6</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 3+2+1=6}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9044f307d28efcac43bd9a515c182878c31685a5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:13.429ex; height:2.343ex;" alt="{\displaystyle 3+2+1=6}"></span>, die zu sich selbst konjugiert sind, deren Ferrers-Diagramm also achsensymmetrisch zu seiner Hauptdiagonalen ist. </p> <ul><li>Die Anzahl der zu sich selbst konjugierten Partitionen von <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span> ist gleich der Anzahl der Partitionen von <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span> in <i>verschiedene, ungerade</i> Summanden.</li></ul> <dl><dd><dl><dd>Beweisidee: Die entscheidende Beobachtung ist, dass jede Spalte im Ferrers-Diagramm, die eine ungerade Anzahl von Kreisen enthält, in der Mitte „gefaltet“ werden kann und so einen Teil eines symmetrischen Diagramms ergibt:</dd></dl></dd></dl> <table> <tbody><tr> <td style="vertical-align:bottom;"><span typeof="mw:File"><a href="/wiki/Datei:GrayDot.svg" class="mw-file-description" title="*"><img alt="*" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/16px-GrayDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/24px-GrayDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/32px-GrayDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span><br /><span typeof="mw:File"><a href="/wiki/Datei:GrayDot.svg" class="mw-file-description" title="*"><img alt="*" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/16px-GrayDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/24px-GrayDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/32px-GrayDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span><br /><span typeof="mw:File"><a href="/wiki/Datei:RedDot.svg" class="mw-file-description" title="*"><img alt="*" src="//upload.wikimedia.org/wikipedia/commons/thumb/e/ec/RedDot.svg/16px-RedDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/ec/RedDot.svg/24px-RedDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/ec/RedDot.svg/32px-RedDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span><br /><span typeof="mw:File"><a href="/wiki/Datei:GrayDot.svg" class="mw-file-description" title="*"><img alt="*" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/16px-GrayDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/24px-GrayDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/32px-GrayDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span><br /><span typeof="mw:File"><a href="/wiki/Datei:GrayDot.svg" class="mw-file-description" title="*"><img alt="*" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/16px-GrayDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/24px-GrayDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/32px-GrayDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span> </td> <td style="vertical-align:middle;">↔ </td> <td style="vertical-align:bottom;"><span typeof="mw:File"><a href="/wiki/Datei:RedDot.svg" class="mw-file-description" title="*"><img alt="*" src="//upload.wikimedia.org/wikipedia/commons/thumb/e/ec/RedDot.svg/16px-RedDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/ec/RedDot.svg/24px-RedDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/ec/RedDot.svg/32px-RedDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span><span typeof="mw:File"><a href="/wiki/Datei:GrayDot.svg" class="mw-file-description" title="*"><img alt="*" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/16px-GrayDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/24px-GrayDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/32px-GrayDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span><span typeof="mw:File"><a href="/wiki/Datei:GrayDot.svg" class="mw-file-description" title="*"><img alt="*" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/16px-GrayDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/24px-GrayDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/32px-GrayDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span><br /><span typeof="mw:File"><a href="/wiki/Datei:GrayDot.svg" class="mw-file-description" title="*"><img alt="*" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/16px-GrayDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/24px-GrayDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/32px-GrayDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span><br /><span typeof="mw:File"><a href="/wiki/Datei:GrayDot.svg" class="mw-file-description" title="*"><img alt="*" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/16px-GrayDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/24px-GrayDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/32px-GrayDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span> </td></tr></tbody></table> <p>Daraus gewinnt man, wie im folgenden Beispiel gezeigt, eine bijektive Abbildung der Partitionen mit verschiedenen, ungeraden Summanden auf die Partitionen, die zu sich selbst konjugiert sind: </p> <table> <tbody><tr> <td valign="top"><span typeof="mw:File"><a href="/wiki/Datei:GrayDot.svg" class="mw-file-description" title="o"><img alt="o" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/16px-GrayDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/24px-GrayDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/32px-GrayDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span><span typeof="mw:File"><a href="/wiki/Datei:RedDot.svg" class="mw-file-description" title="*"><img alt="*" src="//upload.wikimedia.org/wikipedia/commons/thumb/e/ec/RedDot.svg/16px-RedDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/ec/RedDot.svg/24px-RedDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/ec/RedDot.svg/32px-RedDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span><span typeof="mw:File"><a href="/wiki/Datei:BlackDot.svg" class="mw-file-description" title="x"><img alt="x" src="//upload.wikimedia.org/wikipedia/commons/thumb/1/11/BlackDot.svg/16px-BlackDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/11/BlackDot.svg/24px-BlackDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/11/BlackDot.svg/32px-BlackDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span><br /><span typeof="mw:File"><a href="/wiki/Datei:GrayDot.svg" class="mw-file-description" title="o"><img alt="o" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/16px-GrayDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/24px-GrayDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/32px-GrayDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span><span typeof="mw:File"><a href="/wiki/Datei:RedDot.svg" class="mw-file-description" title="*"><img alt="*" src="//upload.wikimedia.org/wikipedia/commons/thumb/e/ec/RedDot.svg/16px-RedDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/ec/RedDot.svg/24px-RedDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/ec/RedDot.svg/32px-RedDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span><span typeof="mw:File"><a href="/wiki/Datei:BlackDot.svg" class="mw-file-description" title="x"><img alt="x" src="//upload.wikimedia.org/wikipedia/commons/thumb/1/11/BlackDot.svg/16px-BlackDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/11/BlackDot.svg/24px-BlackDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/11/BlackDot.svg/32px-BlackDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span><br /><span typeof="mw:File"><a href="/wiki/Datei:GrayDot.svg" class="mw-file-description" title="o"><img alt="o" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/16px-GrayDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/24px-GrayDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/32px-GrayDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span><span typeof="mw:File"><a href="/wiki/Datei:RedDot.svg" class="mw-file-description" title="*"><img alt="*" src="//upload.wikimedia.org/wikipedia/commons/thumb/e/ec/RedDot.svg/16px-RedDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/ec/RedDot.svg/24px-RedDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/ec/RedDot.svg/32px-RedDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span><span typeof="mw:File"><a href="/wiki/Datei:BlackDot.svg" class="mw-file-description" title="x"><img alt="x" src="//upload.wikimedia.org/wikipedia/commons/thumb/1/11/BlackDot.svg/16px-BlackDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/11/BlackDot.svg/24px-BlackDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/11/BlackDot.svg/32px-BlackDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span><br /><span typeof="mw:File"><a href="/wiki/Datei:GrayDot.svg" class="mw-file-description" title="o"><img alt="o" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/16px-GrayDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/24px-GrayDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/32px-GrayDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span><span typeof="mw:File"><a href="/wiki/Datei:RedDot.svg" class="mw-file-description" title="*"><img alt="*" src="//upload.wikimedia.org/wikipedia/commons/thumb/e/ec/RedDot.svg/16px-RedDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/ec/RedDot.svg/24px-RedDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/ec/RedDot.svg/32px-RedDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span><br /><span typeof="mw:File"><a href="/wiki/Datei:GrayDot.svg" class="mw-file-description" title="o"><img alt="o" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/16px-GrayDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/24px-GrayDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/32px-GrayDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span><span typeof="mw:File"><a href="/wiki/Datei:RedDot.svg" class="mw-file-description" title="*"><img alt="*" src="//upload.wikimedia.org/wikipedia/commons/thumb/e/ec/RedDot.svg/16px-RedDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/ec/RedDot.svg/24px-RedDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/ec/RedDot.svg/32px-RedDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span><br /><span typeof="mw:File"><a href="/wiki/Datei:GrayDot.svg" class="mw-file-description" title="o"><img alt="o" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/16px-GrayDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/24px-GrayDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/32px-GrayDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span><span typeof="mw:File"><a href="/wiki/Datei:RedDot.svg" class="mw-file-description" title="*"><img alt="*" src="//upload.wikimedia.org/wikipedia/commons/thumb/e/ec/RedDot.svg/16px-RedDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/ec/RedDot.svg/24px-RedDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/ec/RedDot.svg/32px-RedDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span><br /><span typeof="mw:File"><a href="/wiki/Datei:GrayDot.svg" class="mw-file-description" title="o"><img alt="o" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/16px-GrayDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/24px-GrayDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/32px-GrayDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span><span typeof="mw:File"><a href="/wiki/Datei:RedDot.svg" class="mw-file-description" title="*"><img alt="*" src="//upload.wikimedia.org/wikipedia/commons/thumb/e/ec/RedDot.svg/16px-RedDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/ec/RedDot.svg/24px-RedDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/ec/RedDot.svg/32px-RedDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span><br /><span typeof="mw:File"><a href="/wiki/Datei:GrayDot.svg" class="mw-file-description" title="o"><img alt="o" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/16px-GrayDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/24px-GrayDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/32px-GrayDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span><br /><span typeof="mw:File"><a href="/wiki/Datei:GrayDot.svg" class="mw-file-description" title="o"><img alt="o" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/16px-GrayDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/24px-GrayDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/32px-GrayDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span> </td> <td style="vertical-align:middle;">↔ </td> <td valign="top"><span typeof="mw:File"><a href="/wiki/Datei:GrayDot.svg" class="mw-file-description" title="o"><img alt="o" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/16px-GrayDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/24px-GrayDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/32px-GrayDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span><span typeof="mw:File"><a href="/wiki/Datei:GrayDot.svg" class="mw-file-description" title="o"><img alt="o" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/16px-GrayDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/24px-GrayDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/32px-GrayDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span><span typeof="mw:File"><a href="/wiki/Datei:GrayDot.svg" class="mw-file-description" title="o"><img alt="o" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/16px-GrayDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/24px-GrayDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/32px-GrayDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span><span typeof="mw:File"><a href="/wiki/Datei:GrayDot.svg" class="mw-file-description" title="o"><img alt="o" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/16px-GrayDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/24px-GrayDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/32px-GrayDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span><span typeof="mw:File"><a href="/wiki/Datei:GrayDot.svg" class="mw-file-description" title="o"><img alt="o" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/16px-GrayDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/24px-GrayDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/32px-GrayDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span><br /><span typeof="mw:File"><a href="/wiki/Datei:GrayDot.svg" class="mw-file-description" title="o"><img alt="o" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/16px-GrayDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/24px-GrayDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/32px-GrayDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span><span typeof="mw:File"><a href="/wiki/Datei:RedDot.svg" class="mw-file-description" title="*"><img alt="*" src="//upload.wikimedia.org/wikipedia/commons/thumb/e/ec/RedDot.svg/16px-RedDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/ec/RedDot.svg/24px-RedDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/ec/RedDot.svg/32px-RedDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span><span typeof="mw:File"><a href="/wiki/Datei:RedDot.svg" class="mw-file-description" title="*"><img alt="*" src="//upload.wikimedia.org/wikipedia/commons/thumb/e/ec/RedDot.svg/16px-RedDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/ec/RedDot.svg/24px-RedDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/ec/RedDot.svg/32px-RedDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span><span typeof="mw:File"><a href="/wiki/Datei:RedDot.svg" class="mw-file-description" title="*"><img alt="*" src="//upload.wikimedia.org/wikipedia/commons/thumb/e/ec/RedDot.svg/16px-RedDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/ec/RedDot.svg/24px-RedDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/ec/RedDot.svg/32px-RedDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span><span typeof="mw:File"><a href="/wiki/Datei:RedDot.svg" class="mw-file-description" title="*"><img alt="*" src="//upload.wikimedia.org/wikipedia/commons/thumb/e/ec/RedDot.svg/16px-RedDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/ec/RedDot.svg/24px-RedDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/ec/RedDot.svg/32px-RedDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span><br /><span typeof="mw:File"><a href="/wiki/Datei:GrayDot.svg" class="mw-file-description" title="o"><img alt="o" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/16px-GrayDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/24px-GrayDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/32px-GrayDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span><span typeof="mw:File"><a href="/wiki/Datei:RedDot.svg" class="mw-file-description" title="*"><img alt="*" src="//upload.wikimedia.org/wikipedia/commons/thumb/e/ec/RedDot.svg/16px-RedDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/ec/RedDot.svg/24px-RedDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/ec/RedDot.svg/32px-RedDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span><span typeof="mw:File"><a href="/wiki/Datei:BlackDot.svg" class="mw-file-description" title="x"><img alt="x" src="//upload.wikimedia.org/wikipedia/commons/thumb/1/11/BlackDot.svg/16px-BlackDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/11/BlackDot.svg/24px-BlackDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/11/BlackDot.svg/32px-BlackDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span><span typeof="mw:File"><a href="/wiki/Datei:BlackDot.svg" class="mw-file-description" title="x"><img alt="x" src="//upload.wikimedia.org/wikipedia/commons/thumb/1/11/BlackDot.svg/16px-BlackDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/11/BlackDot.svg/24px-BlackDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/11/BlackDot.svg/32px-BlackDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span><br /><span typeof="mw:File"><a href="/wiki/Datei:GrayDot.svg" class="mw-file-description" title="o"><img alt="o" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/16px-GrayDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/24px-GrayDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/32px-GrayDot.svg.png 2x" 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//upload.wikimedia.org/wikipedia/commons/thumb/1/11/BlackDot.svg/32px-BlackDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span><br /><span typeof="mw:File"><a href="/wiki/Datei:GrayDot.svg" class="mw-file-description" title="o"><img alt="o" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/16px-GrayDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/24px-GrayDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/32px-GrayDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span><span typeof="mw:File"><a href="/wiki/Datei:RedDot.svg" class="mw-file-description" title="*"><img alt="*" src="//upload.wikimedia.org/wikipedia/commons/thumb/e/ec/RedDot.svg/16px-RedDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/ec/RedDot.svg/24px-RedDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/ec/RedDot.svg/32px-RedDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span> </td></tr> <tr style="vertical-align:top; text-align:center;"> <td>9 + 7 + 3 </td> <td>= </td> <td>5 + 5 + 4 + 3 + 2 </td></tr> </tbody></table> <p>Mit ähnlichen Methoden können zum Beispiel die folgenden Aussagen bewiesen werden: Die Anzahl der Partitionen von <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span> mit höchstens <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}"></span> Summanden ist gleich </p> <ol><li>der Anzahl der Partitionen von <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span>, bei denen kein Summand größer als <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}"></span> ist.</li> <li>der Anzahl der Partitionen von <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n+k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>+</mo> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n+k}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/94ccf8f07ceddde4d06fab179c36ccd2c264c243" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.446ex; height:2.343ex;" alt="{\displaystyle n+k}"></span> mit genau <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}"></span> Summanden.</li></ol> <div class="mw-heading mw-heading3"><h3 id="Formalisierung">Formalisierung</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Partitionsfunktion&veaction=edit&section=14" title="Abschnitt bearbeiten: Formalisierung" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Partitionsfunktion&action=edit&section=14" title="Quellcode des Abschnitts bearbeiten: Formalisierung"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Die Ferrers-Diagramme sind ein intuitives Hilfsmittel, mit denen sich Zusammenhänge zwischen ungeordneten Partitionen anschaulich erkennen und nachvollziehen lassen. Für die Erzeugung mit Computern und kompakte Speicherung sind sie ungeeignet, daher spielen auch „formalisierte“ Repräsentationen für diese Diagramme eine wichtige Rolle: </p> <ol><li>Eine Zahlpartition von <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\in \mathbb {N} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>∈<!-- ∈ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\in \mathbb {N} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d059936e77a2d707e9ee0a1d9575a1d693ce5d0b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.913ex; height:2.176ex;" alt="{\displaystyle n\in \mathbb {N} }"></span> („Diagramm der Ordnung <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span>“) ist ein <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fc55753007cd3c18576f7933f6f089196732029" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}"></span>-Tupel („Anzahl der Spalten=Columns“) <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi =\operatorname {(i_{k})} _{k=1}^{C}\in \left(\mathbb {N} \setminus \{0\}\right)^{C}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>π<!-- π --></mi> <mo>=</mo> <msubsup> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mo stretchy="false">(</mo> <msub> <mi mathvariant="normal">i</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">k</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>C</mi> </mrow> </msubsup> <mo>∈<!-- ∈ --></mo> <msup> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> <mo class="MJX-variant">∖<!-- ∖ --></mo> <mo fence="false" stretchy="false">{</mo> <mn>0</mn> <mo fence="false" stretchy="false">}</mo> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>C</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi =\operatorname {(i_{k})} _{k=1}^{C}\in \left(\mathbb {N} \setminus \{0\}\right)^{C}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3cde5f545745d2dcc6f6517d17060b6e99f4a374" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:24.667ex; height:3.509ex;" alt="{\displaystyle \pi =\operatorname {(i_{k})} _{k=1}^{C}\in \left(\mathbb {N} \setminus \{0\}\right)^{C}}"></span> mit der Eigenschaft <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum \nolimits _{k=1}^{C}i_{k}=n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mo movablelimits="false">∑<!-- ∑ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>C</mi> </mrow> </msubsup> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>=</mo> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum \nolimits _{k=1}^{C}i_{k}=n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/64108a7e6c93870aa053312618f6f9032afa10d4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:13.316ex; height:4.343ex;" alt="{\displaystyle \sum \nolimits _{k=1}^{C}i_{k}=n}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fc55753007cd3c18576f7933f6f089196732029" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}"></span> heißt ihre <i>Spaltenzahl</i>. (Um hier auch die „leere“ Partition <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ()=\operatorname {()} _{k=1}^{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mo stretchy="false">)</mo> <mo>=</mo> <msubsup> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mo stretchy="false">(</mo> <mo stretchy="false">)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ()=\operatorname {()} _{k=1}^{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8439478ad25cc4e01f007f155bd15aa394e5f0ac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:9.906ex; height:3.509ex;" alt="{\displaystyle ()=\operatorname {()} _{k=1}^{0}}"></span> mitzuerfassen, muss man für <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f750a7094a396d89a81974cdf35783db2bb287b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.027ex; height:2.176ex;" alt="{\displaystyle C=0}"></span> setzen <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left(\mathbb {N} \setminus \{0\}\right)^{0}:=\{()\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> <mo class="MJX-variant">∖<!-- ∖ --></mo> <mo fence="false" stretchy="false">{</mo> <mn>0</mn> <mo fence="false" stretchy="false">}</mo> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msup> <mo>:=</mo> <mo fence="false" stretchy="false">{</mo> <mo stretchy="false">(</mo> <mo stretchy="false">)</mo> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left(\mathbb {N} \setminus \{0\}\right)^{0}:=\{()\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c0eed16b0f3074e69f12f4b9cdf668dda73e4274" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.103ex; height:3.343ex;" alt="{\displaystyle \left(\mathbb {N} \setminus \{0\}\right)^{0}:=\{()\}}"></span>, es ist dann die <a href="/wiki/Leere_Summe" title="Leere Summe">leere Summe</a> und ergibt immer 0.)</li> <li>Die Zahl <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R=R(\pi )=\max(0,\max\{i_{k}:1\leq k\leq C\})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <mo>=</mo> <mi>R</mi> <mo stretchy="false">(</mo> <mi>π<!-- π --></mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo movablelimits="true" form="prefix">max</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mo movablelimits="true" form="prefix">max</mo> <mo fence="false" stretchy="false">{</mo> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>:</mo> <mn>1</mn> <mo>≤<!-- ≤ --></mo> <mi>k</mi> <mo>≤<!-- ≤ --></mo> <mi>C</mi> <mo fence="false" stretchy="false">}</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R=R(\pi )=\max(0,\max\{i_{k}:1\leq k\leq C\})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59b58775d2101a72b3692708db79cf0a5b77600a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:42.014ex; height:2.843ex;" alt="{\displaystyle R=R(\pi )=\max(0,\max\{i_{k}:1\leq k\leq C\})}"></span> heißt die <i>Zeilenzahl</i> (=„Rows“) von <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi =\operatorname {(i_{k})} _{k=1}^{C}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>π<!-- π --></mi> <mo>=</mo> <msubsup> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mo stretchy="false">(</mo> <msub> <mi mathvariant="normal">i</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">k</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>C</mi> </mrow> </msubsup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi =\operatorname {(i_{k})} _{k=1}^{C}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3394cf2b3e7476974933fbc1893d39c675e3ad2b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:11.823ex; height:3.509ex;" alt="{\displaystyle \pi =\operatorname {(i_{k})} _{k=1}^{C}.}"></span></li> <li>Eine Zahlpartition <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi =\operatorname {(i_{k})} _{k=1}^{C}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>π<!-- π --></mi> <mo>=</mo> <msubsup> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mo stretchy="false">(</mo> <msub> <mi mathvariant="normal">i</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">k</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>C</mi> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi =\operatorname {(i_{k})} _{k=1}^{C}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ee5117d05a90eff902ba590cd92899002d2754ed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:11.176ex; height:3.509ex;" alt="{\displaystyle \pi =\operatorname {(i_{k})} _{k=1}^{C}}"></span> heißt „gültig“, wenn für <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1\leq k<C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>≤<!-- ≤ --></mo> <mi>k</mi> <mo><</mo> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1\leq k<C}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8df8e6c8d839dcfed2a323fd28d76a3667a62039" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:10.337ex; height:2.343ex;" alt="{\displaystyle 1\leq k<C}"></span> stets <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i_{k}\geq i_{k+1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>≥<!-- ≥ --></mo> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i_{k}\geq i_{k+1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5b5683d5b7bc1da7a98d86e2e299b26d1605fef7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.981ex; height:2.509ex;" alt="{\displaystyle i_{k}\geq i_{k+1}}"></span> gilt, für gültige Partitionen mit <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C>0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> <mo>></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C>0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c84d4126c6df243734f9355927c026df6b0d3859" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.027ex; height:2.176ex;" alt="{\displaystyle C>0}"></span> ist <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R(\pi )=i_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <mo stretchy="false">(</mo> <mi>π<!-- π --></mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R(\pi )=i_{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5671a96bb51c834cf02b9a200b7016a4c657eb60" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.86ex; height:2.843ex;" alt="{\displaystyle R(\pi )=i_{1}}"></span>.</li> <li>Eine Zahlpartition <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi =\operatorname {(i_{k})} _{k=1}^{C}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>π<!-- π --></mi> <mo>=</mo> <msubsup> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mo stretchy="false">(</mo> <msub> <mi mathvariant="normal">i</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">k</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>C</mi> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi =\operatorname {(i_{k})} _{k=1}^{C}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ee5117d05a90eff902ba590cd92899002d2754ed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:11.176ex; height:3.509ex;" alt="{\displaystyle \pi =\operatorname {(i_{k})} _{k=1}^{C}}"></span> heißt „strikt“, wenn für <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1\leq k<C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>≤<!-- ≤ --></mo> <mi>k</mi> <mo><</mo> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1\leq k<C}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8df8e6c8d839dcfed2a323fd28d76a3667a62039" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:10.337ex; height:2.343ex;" alt="{\displaystyle 1\leq k<C}"></span> stets <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i_{k}>i_{k+1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>></mo> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i_{k}>i_{k+1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d8032455d339648f8273c961be0f711fbc7116ae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.981ex; height:2.509ex;" alt="{\displaystyle i_{k}>i_{k+1}}"></span> gilt. Strikte Partitionen sind immer gültig.</li> <li>Die <i>konjugierte Partition</i> einer gültigen Partition <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi =\operatorname {(i_{k})} _{k=1}^{C}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>π<!-- π --></mi> <mo>=</mo> <msubsup> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mo stretchy="false">(</mo> <msub> <mi mathvariant="normal">i</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">k</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>C</mi> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi =\operatorname {(i_{k})} _{k=1}^{C}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ee5117d05a90eff902ba590cd92899002d2754ed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:11.176ex; height:3.509ex;" alt="{\displaystyle \pi =\operatorname {(i_{k})} _{k=1}^{C}}"></span> ist definiert durch <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overline {\pi }}=\operatorname {\left(\max\{k:\,{i_{k}}+1-r>0\land 1\leq k\leq C\}\right)} _{r=1}^{R(\pi )}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>π<!-- π --></mi> <mo accent="false">¯<!-- ¯ --></mo> </mover> </mrow> <mo>=</mo> <msubsup> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mrow> <mo>(</mo> <mrow> <mo movablelimits="true" form="prefix">max</mo> <mo fence="false" stretchy="false">{</mo> <mi mathvariant="normal">k</mi> <mo>:</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi mathvariant="normal">i</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">k</mi> </mrow> </msub> </mrow> <mo>+</mo> <mn>1</mn> <mtext>-</mtext> <mi mathvariant="normal">r</mi> <mo>></mo> <mn>0</mn> <mo>∧<!-- ∧ --></mo> <mn>1</mn> <mo>≤<!-- ≤ --></mo> <mi mathvariant="normal">k</mi> <mo>≤<!-- ≤ --></mo> <mi mathvariant="normal">C</mi> <mo fence="false" stretchy="false">}</mo> </mrow> <mo>)</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>R</mi> <mo stretchy="false">(</mo> <mi>π<!-- π --></mi> <mo stretchy="false">)</mo> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {\pi }}=\operatorname {\left(\max\{k:\,{i_{k}}+1-r>0\land 1\leq k\leq C\}\right)} _{r=1}^{R(\pi )}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3c63ae91d66411e500fdb677c34207c1a5034a5b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:44.807ex; height:3.676ex;" alt="{\displaystyle {\overline {\pi }}=\operatorname {\left(\max\{k:\,{i_{k}}+1-r>0\land 1\leq k\leq C\}\right)} _{r=1}^{R(\pi )}}"></span>. Sie ist gültig.</li></ol> <p>Alternativ und näher an der grafischen Darstellung der Ferrers-Diagramme kann man jede Partition als <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R\times C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <mo>×<!-- × --></mo> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R\times C}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2ab06de64d12d83404810974e5d778bb9ca29041" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.371ex; height:2.176ex;" alt="{\displaystyle R\times C}"></span>-<a href="/wiki/Matrix_(Mathematik)" title="Matrix (Mathematik)">Matrix</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (a_{jk})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mi>k</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (a_{jk})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/551f64cb5e923439d51befcd658cb531222b13e7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:4.805ex; height:3.009ex;" alt="{\displaystyle (a_{jk})}"></span> mit Einträgen aus <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{0,1\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{0,1\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/28de5781698336d21c9c560fb1cbb3fb406923eb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.684ex; height:2.843ex;" alt="{\displaystyle \{0,1\}}"></span> darstellen, wobei <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{jk}=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mi>k</mi> </mrow> </msub> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{jk}=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0f421a3652bd74b4218c7fa973d49ff9d150254f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.257ex; height:2.843ex;" alt="{\displaystyle a_{jk}=1}"></span> bedeutet, dass sich im Ferrers-Diagramm in der Reihe <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle j}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>j</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle j}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2f461e54f5c093e92a55547b9764291390f0b5d0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.027ex; width:0.985ex; height:2.509ex;" alt="{\displaystyle j}"></span> in Spalte <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}"></span> ein Kreis befindet, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{jk}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mi>k</mi> </mrow> </msub> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{jk}=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/62b956e87bf6d969bd9c61e4289283a7b54053bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.257ex; height:2.843ex;" alt="{\displaystyle a_{jk}=0}"></span>, dass dort kein Kreis ist. Die Konjugierte einer Partition hat dann als Matrix die <a href="/wiki/Transponierte_Matrix" title="Transponierte Matrix">transponierte Matrix</a> der ursprünglichen Partition. </p> <div class="mw-heading mw-heading2"><h2 id="Varianten">Varianten</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Partitionsfunktion&veaction=edit&section=15" title="Abschnitt bearbeiten: Varianten" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Partitionsfunktion&action=edit&section=15" title="Quellcode des Abschnitts bearbeiten: Varianten"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Partitionen_mit_vorgegebenem_kleinstem_Summanden,_p(k,n)"><span id="Partitionen_mit_vorgegebenem_kleinstem_Summanden.2C_p.28k.2Cn.29"></span>Partitionen mit vorgegebenem kleinstem Summanden, p(k,n)</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Partitionsfunktion&veaction=edit&section=16" title="Abschnitt bearbeiten: Partitionen mit vorgegebenem kleinstem Summanden, p(k,n)" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Partitionsfunktion&action=edit&section=16" title="Quellcode des Abschnitts bearbeiten: Partitionen mit vorgegebenem kleinstem Summanden, p(k,n)"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Bei einer Abwandlung der Partitionsfunktion wird verlangt, dass der kleinste Summand in der Zahlpartition größer oder gleich <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}"></span> ist. Die Anzahl solcher Partitionen wird als <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p(k,n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo stretchy="false">(</mo> <mi>k</mi> <mo>,</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p(k,n)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cce5eb8ef4b282b99cd1e393c595ae26cc30e776" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:6.708ex; height:2.843ex;" alt="{\displaystyle p(k,n)}"></span> notiert. Die „normale“ Partitionsfunktion ist somit <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(n)=p(1,n).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>p</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(n)=p(1,n).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1ab6beb8e2b18b26418556502bdfb88d0fcedce0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.265ex; height:2.843ex;" alt="{\displaystyle P(n)=p(1,n).}"></span> Diese Abwandlung <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p(k,n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo stretchy="false">(</mo> <mi>k</mi> <mo>,</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p(k,n)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cce5eb8ef4b282b99cd1e393c595ae26cc30e776" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:6.708ex; height:2.843ex;" alt="{\displaystyle p(k,n)}"></span> ist Folge <a href="//oeis.org/A026807" class="extiw" title="oeis:A026807">A026807</a> in <a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>. </p><p><b>Beispielwerte für p(k,n)</b> </p> <table class="wikitable float-right"> <caption>Beispielwerte von p(k,n) </caption> <tbody><tr class="hintergrundfarbe5"> <td colspan="2" rowspan="2"><b>p(k,n)</b></td> <td colspan="10" style="text-align:center"><b>k</b> </td></tr> <tr align="right"> <th>1</th> <th>2</th> <th>3</th> <th>4</th> <th>5</th> <th>6</th> <th>7</th> <th>8</th> <th>9</th> <th>10 </th></tr> <tr align="right"> <td rowspan="10" class="hintergrundfarbe5" align="center"><b>n</b></td> <td><b>1</b></td> <td>1 </td></tr> <tr align="right"> <td><b>2</b></td> <td>2</td> <td>1 </td></tr> <tr align="right"> <td><b>3</b></td> <td>3</td> <td>1</td> <td>1 </td></tr> <tr align="right"> <td><b>4</b></td> <td>5</td> <td>2</td> <td>1</td> <td>1 </td></tr> <tr align="right"> <td><b>5</b></td> <td>7</td> <td>2</td> <td>1</td> <td>1</td> <td>1 </td></tr> <tr align="right"> <td><b>6</b></td> <td>11</td> <td>4</td> <td>2</td> <td>1</td> <td>1</td> <td>1 </td></tr> <tr align="right"> <td><b>7</b></td> <td>15</td> <td>4</td> <td>2</td> <td>1</td> <td>1</td> <td>1</td> <td>1 </td></tr> <tr align="right"> <td><b>8</b></td> <td>22</td> <td>7</td> <td>3</td> <td>2</td> <td>1</td> <td>1</td> <td>1</td> <td>1 </td></tr> <tr align="right"> <td><b>9</b></td> <td>30</td> <td>8</td> <td>4</td> <td>2</td> <td>1</td> <td>1</td> <td>1</td> <td>1</td> <td>1 </td></tr> <tr align="right"> <td><b>10</b></td> <td>42</td> <td>12</td> <td>5</td> <td>3</td> <td>2</td> <td>1</td> <td>1</td> <td>1</td> <td>1</td> <td>1 </td></tr></tbody></table> <p>Zu den Werten von <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p(k,n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo stretchy="false">(</mo> <mi>k</mi> <mo>,</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p(k,n)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cce5eb8ef4b282b99cd1e393c595ae26cc30e776" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:6.708ex; height:2.843ex;" alt="{\displaystyle p(k,n)}"></span> für kleine Zahlen siehe auch die zweite Tabelle rechts. Einzelwerte sind: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{alignedat}{4}p(1,4)&=5,&p(2,8)&=7,&p(3,12)&=9,\\p(4,16)&=11,\quad &p(5,20)&=13,\quad &p(6,24)&=16.\end{alignedat}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left" rowspacing="3pt" columnspacing="0em 0em 0em 0em 0em 0em 0em 0em" displaystyle="true"> <mtr> <mtd> <mi>p</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mn>4</mn> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mn>5</mn> <mo>,</mo> </mtd> <mtd> <mi>p</mi> <mo stretchy="false">(</mo> <mn>2</mn> <mo>,</mo> <mn>8</mn> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mn>7</mn> <mo>,</mo> </mtd> <mtd> <mi>p</mi> <mo stretchy="false">(</mo> <mn>3</mn> <mo>,</mo> <mn>12</mn> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mn>9</mn> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mi>p</mi> <mo stretchy="false">(</mo> <mn>4</mn> <mo>,</mo> <mn>16</mn> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mn>11</mn> <mo>,</mo> <mspace width="1em" /> </mtd> <mtd> <mi>p</mi> <mo stretchy="false">(</mo> <mn>5</mn> <mo>,</mo> <mn>20</mn> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mn>13</mn> <mo>,</mo> <mspace width="1em" /> </mtd> <mtd> <mi>p</mi> <mo stretchy="false">(</mo> <mn>6</mn> <mo>,</mo> <mn>24</mn> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mn>16.</mn> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{alignedat}{4}p(1,4)&=5,&p(2,8)&=7,&p(3,12)&=9,\\p(4,16)&=11,\quad &p(5,20)&=13,\quad &p(6,24)&=16.\end{alignedat}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/37e623f98bb19fb1a0b503ccc1686ba67d267237" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:46.107ex; height:6.176ex;" alt="{\displaystyle {\begin{alignedat}{4}p(1,4)&=5,&p(2,8)&=7,&p(3,12)&=9,\\p(4,16)&=11,\quad &p(5,20)&=13,\quad &p(6,24)&=16.\end{alignedat}}}"></span></dd></dl> <div class="mw-heading mw-heading4"><h4 id="Rekursionsformel_für_p(k,n)_und_P(n)"><span id="Rekursionsformel_f.C3.BCr_p.28k.2Cn.29_und_P.28n.29"></span>Rekursionsformel für p(k,n) und P(n)</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Partitionsfunktion&veaction=edit&section=17" title="Abschnitt bearbeiten: Rekursionsformel für p(k,n) und P(n)" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Partitionsfunktion&action=edit&section=17" title="Quellcode des Abschnitts bearbeiten: Rekursionsformel für p(k,n) und P(n)"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Es gilt </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p(n,n)=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>,</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p(n,n)=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/569cdc60497b8f912b9cd8a05d85c532b7028da3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:11.152ex; height:2.843ex;" alt="{\displaystyle p(n,n)=1}"></span></dd></dl> <p>und </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p(k,n)=1+\sum _{j=k}^{\left\lfloor {\frac {n}{2}}\right\rfloor }p(j,n-j)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo stretchy="false">(</mo> <mi>k</mi> <mo>,</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1</mn> <mo>+</mo> <munderover> <mo>∑<!-- ∑ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>=</mo> <mi>k</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>⌊</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>n</mi> <mn>2</mn> </mfrac> </mrow> <mo>⌋</mo> </mrow> </mrow> </munderover> <mi>p</mi> <mo stretchy="false">(</mo> <mi>j</mi> <mo>,</mo> <mi>n</mi> <mo>−<!-- − --></mo> <mi>j</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p(k,n)=1+\sum _{j=k}^{\left\lfloor {\frac {n}{2}}\right\rfloor }p(j,n-j)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/179984f2a5f2ab53b822296fbb8e06d92e4ce550" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; margin-left: -0.089ex; width:27.733ex; height:9.176ex;" alt="{\displaystyle p(k,n)=1+\sum _{j=k}^{\left\lfloor {\frac {n}{2}}\right\rfloor }p(j,n-j)}"></span></dd></dl> <p>für <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k<n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo><</mo> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k<n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/48ab7000a6f47e3a09a79dcbe31b89272b0c1f1f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.704ex; height:2.176ex;" alt="{\displaystyle k<n}"></span>, wobei <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lfloor n\rfloor }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⌊<!-- ⌊ --></mo> <mi>n</mi> <mo fence="false" stretchy="false">⌋<!-- ⌋ --></mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lfloor n\rfloor }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bbedac79c33978cc2ddb110806bff07edf3157b4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.459ex; height:2.843ex;" alt="{\displaystyle \lfloor n\rfloor }"></span> die <a href="/wiki/Gau%C3%9Fklammer" class="mw-redirect" title="Gaußklammer">Gaußklammer</a> ist. Mit dieser <i>Rekursionsformel</i> lassen sich alle Werte von <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p(k,n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo stretchy="false">(</mo> <mi>k</mi> <mo>,</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p(k,n)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cce5eb8ef4b282b99cd1e393c595ae26cc30e776" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:6.708ex; height:2.843ex;" alt="{\displaystyle p(k,n)}"></span> und damit auch für <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(n)=p(1,n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>p</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(n)=p(1,n)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4f7d64e649672bc62843d215c43c97fee3a76da4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.618ex; height:2.843ex;" alt="{\displaystyle P(n)=p(1,n)}"></span> berechnen. Man beachte aber, dass bei der Rekursionsformel für die Berechnung von <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p(1,N)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mi>N</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p(1,N)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3f6d9883dd02eac8a46ec6d7360d9d67ece0f3be" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:7.328ex; height:2.843ex;" alt="{\displaystyle p(1,N)}"></span> alle Werte von <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p(k,n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo stretchy="false">(</mo> <mi>k</mi> <mo>,</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p(k,n)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cce5eb8ef4b282b99cd1e393c595ae26cc30e776" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:6.708ex; height:2.843ex;" alt="{\displaystyle p(k,n)}"></span> für <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1<n<N}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo><</mo> <mi>n</mi> <mo><</mo> <mi>N</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1<n<N}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/40aa7808fb9cb885b4185bd47c8b6d0df906e647" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.818ex; height:2.176ex;" alt="{\displaystyle 1<n<N}"></span> und <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1\leq k<\left\lfloor {\frac {N}{2}}\right\rfloor }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>≤<!-- ≤ --></mo> <mi>k</mi> <mo><</mo> <mrow> <mo>⌊</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>N</mi> <mn>2</mn> </mfrac> </mrow> <mo>⌋</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1\leq k<\left\lfloor {\frac {N}{2}}\right\rfloor }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a4f448669d046e8b7dca71ebd4ce5f99f5e4889d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:14.181ex; height:6.176ex;" alt="{\displaystyle 1\leq k<\left\lfloor {\frac {N}{2}}\right\rfloor }"></span> bekannt sein oder mitberechnet werden müssen. </p> <div class="mw-heading mw-heading3"><h3 id="Geordnete_Zahlpartitionen">Geordnete Zahlpartitionen</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Partitionsfunktion&veaction=edit&section=18" title="Abschnitt bearbeiten: Geordnete Zahlpartitionen" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Partitionsfunktion&action=edit&section=18" title="Quellcode des Abschnitts bearbeiten: Geordnete Zahlpartitionen"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Betrachtet man die Summanden in einer Zahlpartition als geordnete Menge, berücksichtigt also die Reihenfolge in der Summe, dann spricht man von einer <i>geordneten Zahlpartition</i>. Hier werden die folgenden Anzahlfunktionen betrachtet, für die kein Formelzeichen allgemein verbreitet ist. </p> <ul><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g(k,n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mo stretchy="false">(</mo> <mi>k</mi> <mo>,</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g(k,n)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e6e0697b390a9c90ac1bc5d45cad605c94c41cfd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.565ex; height:2.843ex;" alt="{\displaystyle g(k,n)}"></span> ist die Anzahl der Darstellungen von <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span> als Summe von genau <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}"></span> positiven ganzen Zahlen mit Berücksichtigung der Reihenfolge der Summanden, also die Anzahl der Lösungen <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (i_{1},i_{2},\ldots ,i_{k})\in \left(\mathbb {N} \setminus \{0\}\right)^{k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>…<!-- … --></mo> <mo>,</mo> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>∈<!-- ∈ --></mo> <msup> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> <mo class="MJX-variant">∖<!-- ∖ --></mo> <mo fence="false" stretchy="false">{</mo> <mn>0</mn> <mo fence="false" stretchy="false">}</mo> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (i_{1},i_{2},\ldots ,i_{k})\in \left(\mathbb {N} \setminus \{0\}\right)^{k}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1faf4270b098e9ee6b60fad7e264bf899938960b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.725ex; height:3.343ex;" alt="{\displaystyle (i_{1},i_{2},\ldots ,i_{k})\in \left(\mathbb {N} \setminus \{0\}\right)^{k}}"></span> der Gleichung <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i_{1}+i_{2}+\cdots +i_{k}=n.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <mo>⋯<!-- ⋯ --></mo> <mo>+</mo> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>=</mo> <mi>n</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i_{1}+i_{2}+\cdots +i_{k}=n.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6f1b42784cddeabf0b2704d3f9445d4b59c67f07" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:21.989ex; height:2.509ex;" alt="{\displaystyle i_{1}+i_{2}+\cdots +i_{k}=n.}"></span></li></ul> <dl><dd><ul><li>Es gilt <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g(k,n)={\binom {n-1}{k-1}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mo stretchy="false">(</mo> <mi>k</mi> <mo>,</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mrow> <mi>n</mi> <mo>−<!-- − --></mo> <mn>1</mn> </mrow> <mrow> <mi>k</mi> <mo>−<!-- − --></mo> <mn>1</mn> </mrow> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g(k,n)={\binom {n-1}{k-1}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/765f23f4b4dcfd28a55c48b74ffe099b7d588cc1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:18.482ex; height:6.176ex;" alt="{\displaystyle g(k,n)={\binom {n-1}{k-1}}}"></span>.<sup id="cite_ref-MaNes_2-3" class="reference"><a href="#cite_note-MaNes-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup></li> <li>Die Anzahl lässt sich geometrisch deuten als Zahl der Punkte mit positiven, ganzzahligen Koordinaten auf der Hyperebene mit der Gleichung <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{1}+x_{2}+\cdots +x_{k}=n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <mo>⋯<!-- ⋯ --></mo> <mo>+</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>=</mo> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{1}+x_{2}+\cdots +x_{k}=n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6ab037e846c32830a1066ad20d92b94c50d9fa4f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:22.924ex; height:2.343ex;" alt="{\displaystyle x_{1}+x_{2}+\cdots +x_{k}=n}"></span> im <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}"></span>-<a href="/wiki/Dimension_(Mathematik)" title="Dimension (Mathematik)">dimensionalen</a> reellen <a href="/wiki/Affiner_Raum" title="Affiner Raum">affinen Punktraum</a>.</li> <li>Die Folge <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g(1,1),g(1,2),g(2,2),g(1,3),g(2,3),g(3,3),g(1,4),\ldots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>,</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo>,</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mn>2</mn> <mo>,</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo>,</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mn>3</mn> <mo stretchy="false">)</mo> <mo>,</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mn>2</mn> <mo>,</mo> <mn>3</mn> <mo stretchy="false">)</mo> <mo>,</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mn>3</mn> <mo>,</mo> <mn>3</mn> <mo stretchy="false">)</mo> <mo>,</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mn>4</mn> <mo stretchy="false">)</mo> <mo>,</mo> <mo>…<!-- … --></mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g(1,1),g(1,2),g(2,2),g(1,3),g(2,3),g(3,3),g(1,4),\ldots }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ff55a6f5beaffd1f8ffe67d6dbdaf6f6268f1d65" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:53.95ex; height:2.843ex;" alt="{\displaystyle g(1,1),g(1,2),g(2,2),g(1,3),g(2,3),g(3,3),g(1,4),\ldots }"></span> ist die Folge der Zahlen im <a href="/wiki/Pascalsches_Dreieck" title="Pascalsches Dreieck">pascalschen Dreieck</a>, den Reihen nach gelesen, Folge <a href="//oeis.org/A007318" class="extiw" title="oeis:A007318">A007318</a> in <a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>.</li></ul></dd></dl> <ul><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G(n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G(n)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/63043eb6b913833d5514f63b622cd1e6d242e2a2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.031ex; height:2.843ex;" alt="{\displaystyle G(n)}"></span> ist die Anzahl der Darstellungen von <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span> als Summe von höchstens <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span> positiven ganzen Zahlen mit Berücksichtigung der Reihenfolge der Summanden. Sie ist Folge <a href="//oeis.org/A000079" class="extiw" title="oeis:A000079">A000079</a> in <a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a> und es gilt<sup id="cite_ref-MaNes_2-4" class="reference"><a href="#cite_note-MaNes-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup></li></ul> <dl><dd><ul><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G(n)=\sum _{j=1}^{n}g(j,n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munderover> <mo>∑<!-- ∑ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mi>g</mi> <mo stretchy="false">(</mo> <mi>j</mi> <mo>,</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G(n)=\sum _{j=1}^{n}g(j,n)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7372f07fac3c972dcb0b4adf03ab623f22a99278" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:18.183ex; height:7.176ex;" alt="{\displaystyle G(n)=\sum _{j=1}^{n}g(j,n)}"></span>,</li> <li>die Rekursionsformel <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G(n)=1+\sum _{i_{1}=1}^{n-1}G(n-i_{1}),\,n\geq 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1</mn> <mo>+</mo> <munderover> <mo>∑<!-- ∑ --></mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−<!-- − --></mo> <mn>1</mn> </mrow> </munderover> <mi>G</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−<!-- − --></mo> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>,</mo> <mspace width="thinmathspace" /> <mi>n</mi> <mo>≥<!-- ≥ --></mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G(n)=1+\sum _{i_{1}=1}^{n-1}G(n-i_{1}),\,n\geq 1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f69f47387df62a5facc722bf090f4158977b5de9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:32.823ex; height:7.676ex;" alt="{\displaystyle G(n)=1+\sum _{i_{1}=1}^{n-1}G(n-i_{1}),\,n\geq 1}"></span> und</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G(n)=2^{n-1},n\geq 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−<!-- − --></mo> <mn>1</mn> </mrow> </msup> <mo>,</mo> <mi>n</mi> <mo>≥<!-- ≥ --></mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G(n)=2^{n-1},n\geq 1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4ef5a95b196a726a5f598c4d4c547cf36f50331b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.3ex; height:3.176ex;" alt="{\displaystyle G(n)=2^{n-1},n\geq 1}"></span>, was sich leicht mit <a href="/wiki/Vollst%C3%A4ndige_Induktion" title="Vollständige Induktion">vollständiger Induktion</a> aus der Rekursionsformel beweisen lässt.</li></ul></dd></dl> <p>Offenbar liefert die leicht zu berechnende Funktion <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G(n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G(n)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/63043eb6b913833d5514f63b622cd1e6d242e2a2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.031ex; height:2.843ex;" alt="{\displaystyle G(n)}"></span> eine (sehr grobe) obere Schranke für die Partitionsfunktion:<sup id="cite_ref-MaNes_2-5" class="reference"><a href="#cite_note-MaNes-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(n)<G(n)=2^{n-1},\ n\geq 3.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo><</mo> <mi>G</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−<!-- − --></mo> <mn>1</mn> </mrow> </msup> <mo>,</mo> <mtext> </mtext> <mi>n</mi> <mo>≥<!-- ≥ --></mo> <mn>3.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(n)<G(n)=2^{n-1},\ n\geq 3.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/41e8f396c2e797396200cd630ae1acbbd5101aee" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.576ex; height:3.176ex;" alt="{\displaystyle P(n)<G(n)=2^{n-1},\ n\geq 3.}"></span></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Strikte_Partitionen_und_verwandte_Nebenbedingungen">Strikte Partitionen und verwandte Nebenbedingungen</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Partitionsfunktion&veaction=edit&section=19" title="Abschnitt bearbeiten: Strikte Partitionen und verwandte Nebenbedingungen" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Partitionsfunktion&action=edit&section=19" title="Quellcode des Abschnitts bearbeiten: Strikte Partitionen und verwandte Nebenbedingungen"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Die Zahlpartitionen von <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span>, die aus lauter ungeraden Summanden bestehen, lassen sich bijektiv abbilden auf die strikten Zahlpartitionen, das sind die Zahlpartitionen mit lauter <i>unterschiedlichen Summanden</i>. Diese Tatsache wurde bereits 1748 von <a href="/wiki/Leonhard_Euler" title="Leonhard Euler">Euler</a> nachgewiesen.<sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> Sie ist ein Spezialfall des <i>Satzes von Glaisher</i> der nach <a href="/wiki/James_Whitbread_Lee_Glaisher" title="James Whitbread Lee Glaisher">James Whitbread Lee Glaisher</a> benannt ist: </p> <dl><dd><i>Die Anzahl der Partitionen von <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span>, bei denen kein Summand durch <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d\in \mathbb {N} \setminus \{0,1\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mo>∈<!-- ∈ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> <mo class="MJX-variant">∖<!-- ∖ --></mo> <mo fence="false" stretchy="false">{</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d\in \mathbb {N} \setminus \{0,1\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f0e9a3d86a523d357387c360a0792c50bc05516" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.613ex; height:2.843ex;" alt="{\displaystyle d\in \mathbb {N} \setminus \{0,1\}}"></span> teilbar ist, gleicht der Anzahl der Partitionen von <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span>, in denen keine <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e85ff03cbe0c7341af6b982e47e9f90d235c66ab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.216ex; height:2.176ex;" alt="{\displaystyle d}"></span> übereinstimmenden Summanden vorkommen.</i><sup id="cite_ref-Lehmer_13-0" class="reference"><a href="#cite_note-Lehmer-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup></dd></dl> <p>Damit verwandt ist die folgende Aussage, die nach <a href="/wiki/Leonard_James_Rogers" title="Leonard James Rogers">Leonard James Rogers</a> als <i>Satz von Rogers</i> benannt ist:<sup id="cite_ref-Lehmer_13-1" class="reference"><a href="#cite_note-Lehmer-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><i>Die Anzahl der Partitionen von <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span>, deren Summanden sich um 2 oder mehr unterscheiden, ist der Anzahl der Partitionen von <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span> gleich, bei der alle Summanden bei Division durch 5 den Rest 1 oder 4 lassen.</i></dd></dl> <p>Die Aussage ist Teil der <a href="/wiki/Rogers-Ramanujan-Identit%C3%A4ten" title="Rogers-Ramanujan-Identitäten">Rogers-Ramanujan-Identitäten</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Mathematische_Anwendungen">Mathematische Anwendungen</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Partitionsfunktion&veaction=edit&section=20" title="Abschnitt bearbeiten: Mathematische Anwendungen" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Partitionsfunktion&action=edit&section=20" title="Quellcode des Abschnitts bearbeiten: Mathematische Anwendungen"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="hauptartikel" role="navigation"><span class="hauptartikel-pfeil" title="siehe" aria-hidden="true" role="presentation">→ </span><i><span class="hauptartikel-text">Hauptartikel</span>: <a href="/wiki/Partitionierungsproblem" title="Partitionierungsproblem">Partitionierungsproblem</a></i></div> <p>zu Anwendungen in Technik und Informatik. </p> <div class="mw-heading mw-heading3"><h3 id="Konjugationsklassen_der_symmetrischen_Gruppe">Konjugationsklassen der symmetrischen Gruppe</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Partitionsfunktion&veaction=edit&section=21" title="Abschnitt bearbeiten: Konjugationsklassen der symmetrischen Gruppe" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Partitionsfunktion&action=edit&section=21" title="Quellcode des Abschnitts bearbeiten: Konjugationsklassen der symmetrischen Gruppe"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="hauptartikel" role="navigation"><span class="hauptartikel-pfeil" title="siehe" aria-hidden="true" role="presentation">→ </span><i><span class="hauptartikel-text">Hauptartikel</span>: <a href="/wiki/Young-Tableau" title="Young-Tableau">Young-Tableau</a></i></div> <p>Die Anzahl der <a href="/wiki/Konjugation_(Gruppentheorie)" title="Konjugation (Gruppentheorie)">Konjugationsklassen</a> in der <a href="/wiki/Symmetrische_Gruppe" title="Symmetrische Gruppe">symmetrischen Gruppe</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9f049ac28d4ac8097b625f9d71c1f22b2ebd1bc4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.643ex; height:2.509ex;" alt="{\displaystyle S_{n}}"></span> ist gleich dem Wert <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(n)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5e303d2c14cd399b6f52b468c9fd44a542bed422" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.949ex; height:2.843ex;" alt="{\displaystyle P(n)}"></span> der Partitionsfunktion, denn jede Konjugationsklasse entspricht genau einem <a href="/wiki/Zykeltyp" title="Zykeltyp">Zykeltyp</a> von Permutationen mit einer bestimmten Struktur der Darstellung in <i>disjunkter Zyklenschreibweise</i>. </p><p><b>Beispiele</b> </p> <ul><li>Die Permutation <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (1,2,3)(5,7,8,9)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>3</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mn>5</mn> <mo>,</mo> <mn>7</mn> <mo>,</mo> <mn>8</mn> <mo>,</mo> <mn>9</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (1,2,3)(5,7,8,9)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3ff86f57a8da4d1ba61a03e15c4fc299edaacdc0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.925ex; height:2.843ex;" alt="{\displaystyle (1,2,3)(5,7,8,9)}"></span> gehört als Element der <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S_{9}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>9</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S_{9}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/756e288b8bd7a33fe5d9051bcda74f1f9a661c0a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.479ex; height:2.509ex;" alt="{\displaystyle S_{9}}"></span> zu der Zahlpartition <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1+1+3+4}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>+</mo> <mn>1</mn> <mo>+</mo> <mn>3</mn> <mo>+</mo> <mn>4</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1+1+3+4}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3ee1311748aea2166958826483c7d5bf6574b7e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:13.171ex; height:2.343ex;" alt="{\displaystyle 1+1+3+4}"></span> der Zahl 9, als Element der <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S_{12}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S_{12}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/615b370a1e38388a9e56151f7c4a9c2e9e374eb1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.301ex; height:2.509ex;" alt="{\displaystyle S_{12}}"></span> zur Zahlpartition <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1+1+1+1+1+3+4}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>+</mo> <mn>1</mn> <mo>+</mo> <mn>1</mn> <mo>+</mo> <mn>1</mn> <mo>+</mo> <mn>1</mn> <mo>+</mo> <mn>3</mn> <mo>+</mo> <mn>4</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1+1+1+1+1+3+4}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fc2e0531a09fc507a4929e8059b41114255629b7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:25.18ex; height:2.343ex;" alt="{\displaystyle 1+1+1+1+1+3+4}"></span> von 12. Man beachte, dass Fixelemente der Permutation, die in der Zyklenschreibweise (als „Einerzyklen“) fast immer fortgelassen werden, in der Zahlpartition als Summanden 1 auftauchen. Jedes Element der <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S_{12}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S_{12}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/615b370a1e38388a9e56151f7c4a9c2e9e374eb1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.301ex; height:2.509ex;" alt="{\displaystyle S_{12}}"></span>, das in der disjunkten Zyklenschreibweise aus einem Dreier- und einem Viererzyklus besteht, ist in <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S_{12}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S_{12}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/615b370a1e38388a9e56151f7c4a9c2e9e374eb1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.301ex; height:2.509ex;" alt="{\displaystyle S_{12}}"></span> zu dem oben genannten Element konjugiert, es gibt in diesem Fall <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\binom {12}{4}}\cdot {\binom {8}{3}}\cdot 3!\cdot 2!=332\,640}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mn>12</mn> <mn>4</mn> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <mo>⋅<!-- ⋅ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mn>8</mn> <mn>3</mn> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <mo>⋅<!-- ⋅ --></mo> <mn>3</mn> <mo>!</mo> <mo>⋅<!-- ⋅ --></mo> <mn>2</mn> <mo>!</mo> <mo>=</mo> <mn>332</mn> <mspace width="thinmathspace" /> <mn>640</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\binom {12}{4}}\cdot {\binom {8}{3}}\cdot 3!\cdot 2!=332\,640}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c06eaf93d05bc31a050142bffda99e4ddde5669c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:29.446ex; height:6.176ex;" alt="{\displaystyle {\binom {12}{4}}\cdot {\binom {8}{3}}\cdot 3!\cdot 2!=332\,640}"></span> solche Permutationen.</li> <li>Die Permutation <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (1,2,3)(5,7,6)(10,11)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>3</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mn>5</mn> <mo>,</mo> <mn>7</mn> <mo>,</mo> <mn>6</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mn>10</mn> <mo>,</mo> <mn>11</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (1,2,3)(5,7,6)(10,11)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/44231a3bdeb1b96d768df820dd75f83d165a209f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.222ex; height:2.843ex;" alt="{\displaystyle (1,2,3)(5,7,6)(10,11)}"></span> gehört als Element der <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S_{12}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S_{12}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/615b370a1e38388a9e56151f7c4a9c2e9e374eb1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.301ex; height:2.509ex;" alt="{\displaystyle S_{12}}"></span> zur Zahlpartition <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1+1+1+1+2+3+3}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>+</mo> <mn>1</mn> <mo>+</mo> <mn>1</mn> <mo>+</mo> <mn>1</mn> <mo>+</mo> <mn>2</mn> <mo>+</mo> <mn>3</mn> <mo>+</mo> <mn>3</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1+1+1+1+2+3+3}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3ee021cad82eedb4dec1e24e9b1e39085b473a51" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:25.18ex; height:2.343ex;" alt="{\displaystyle 1+1+1+1+2+3+3}"></span> von 12. Sie gehört in <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S_{12}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S_{12}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/615b370a1e38388a9e56151f7c4a9c2e9e374eb1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.301ex; height:2.509ex;" alt="{\displaystyle S_{12}}"></span> zu einer Konjugationsklasse, die <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\binom {12}{3}}\cdot {\binom {9}{3}}\cdot {\binom {6}{2}}\cdot {\frac {2!\cdot 2!}{2!}}=554\,400}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mn>12</mn> <mn>3</mn> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <mo>⋅<!-- ⋅ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mn>9</mn> <mn>3</mn> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <mo>⋅<!-- ⋅ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mn>6</mn> <mn>2</mn> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <mo>⋅<!-- ⋅ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mo>!</mo> <mo>⋅<!-- ⋅ --></mo> <mn>2</mn> <mo>!</mo> </mrow> <mrow> <mn>2</mn> <mo>!</mo> </mrow> </mfrac> </mrow> <mo>=</mo> <mn>554</mn> <mspace width="thinmathspace" /> <mn>400</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\binom {12}{3}}\cdot {\binom {9}{3}}\cdot {\binom {6}{2}}\cdot {\frac {2!\cdot 2!}{2!}}=554\,400}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a1b650d11dc65f4d90342f00d62e10cf638d4a62" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:36.545ex; height:6.176ex;" alt="{\displaystyle {\binom {12}{3}}\cdot {\binom {9}{3}}\cdot {\binom {6}{2}}\cdot {\frac {2!\cdot 2!}{2!}}=554\,400}"></span> Permutationen enthält.</li></ul> <div class="mw-heading mw-heading3"><h3 id="Zahlpartition_und_endliche_Mengenpartition">Zahlpartition und endliche Mengenpartition</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Partitionsfunktion&veaction=edit&section=22" title="Abschnitt bearbeiten: Zahlpartition und endliche Mengenpartition" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Partitionsfunktion&action=edit&section=22" title="Quellcode des Abschnitts bearbeiten: Zahlpartition und endliche Mengenpartition"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Jede <a href="/wiki/%C3%84quivalenzrelation" title="Äquivalenzrelation">Äquivalenzrelation</a> auf einer endlichen Menge <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}"></span> mit <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span> Elementen bestimmt eine <a href="/wiki/Partition_(Mengenlehre)" title="Partition (Mengenlehre)">Mengenpartition</a> von <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}"></span>. In der Kombinatorik wird ohne Einschränkung der Allgemeinheit <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M=\{1,2,\ldots n\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mo>…<!-- … --></mo> <mi>n</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M=\{1,2,\ldots n\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9c339d97f655424fec7c41ffff9c74610d780fc0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.763ex; height:2.843ex;" alt="{\displaystyle M=\{1,2,\ldots n\}}"></span> angenommen. Zu jeder <i>Zahlpartition</i> von <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span> gehört eine nicht leere Menge von isomorphen Äquivalenzklasseneinteilungen der Menge <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}"></span>. Die Anzahl der Zahlpartitionen von <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span> ist daher kleiner gleich der Anzahl der Mengenpartitionen von <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}"></span>, für <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\geq 3}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>≥<!-- ≥ --></mo> <mn>3</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\geq 3}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/73136e4a27fe39c123d16a7808e76d3162ce42bb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.656ex; height:2.343ex;" alt="{\displaystyle n\geq 3}"></span> echt kleiner: </p> <table class="wikitable float-right"> <caption><b>Beispiele</b> </caption> <tbody><tr> <th><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span> </th> <td>0</td> <td>1</td> <td>2</td> <td>3</td> <td>4</td> <td>5</td> <td>6</td> <td>7</td> <td>8</td> <td>9</td> <td>10</td> <td>11 </td></tr> <tr> <th>Anzahl der Zahlpartitionen <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(n)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5e303d2c14cd399b6f52b468c9fd44a542bed422" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.949ex; height:2.843ex;" alt="{\displaystyle P(n)}"></span> </th> <td>1</td> <td>1</td> <td>2</td> <td>3</td> <td>5</td> <td>7</td> <td>11</td> <td>15</td> <td>22</td> <td>30</td> <td>42</td> <td>56 </td></tr> <tr> <th>Anzahl der Mengenpartitionen <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2f568bf6d34e97b9fdda0dc7e276d6c4501d2045" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.982ex; height:2.509ex;" alt="{\displaystyle B_{n}}"></span> </th> <td>1</td> <td>1</td> <td>2</td> <td>5</td> <td>15</td> <td>52</td> <td>203</td> <td>877</td> <td>4140</td> <td>21147</td> <td>115975</td> <td>678570 </td></tr></tbody></table> <ul><li>Zu der Zahlpartition <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 3=2+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>3</mn> <mo>=</mo> <mn>2</mn> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 3=2+1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/498ca7b6b03b339ee2169342000881daaed64579" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.426ex; height:2.343ex;" alt="{\displaystyle 3=2+1}"></span> von 3 gehören die 3 Mengenpartitionen <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{\{1,2\},\{3\}\};\{\{1,3\},\{2\}\};\{\{2,3\},\{1\}\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mo fence="false" stretchy="false">{</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo fence="false" stretchy="false">}</mo> <mo>,</mo> <mo fence="false" stretchy="false">{</mo> <mn>3</mn> <mo fence="false" stretchy="false">}</mo> <mo fence="false" stretchy="false">}</mo> <mo>;</mo> <mo fence="false" stretchy="false">{</mo> <mo fence="false" stretchy="false">{</mo> <mn>1</mn> <mo>,</mo> <mn>3</mn> <mo fence="false" stretchy="false">}</mo> <mo>,</mo> <mo fence="false" stretchy="false">{</mo> <mn>2</mn> <mo fence="false" stretchy="false">}</mo> <mo fence="false" stretchy="false">}</mo> <mo>;</mo> <mo fence="false" stretchy="false">{</mo> <mo fence="false" stretchy="false">{</mo> <mn>2</mn> <mo>,</mo> <mn>3</mn> <mo fence="false" stretchy="false">}</mo> <mo>,</mo> <mo fence="false" stretchy="false">{</mo> <mn>1</mn> <mo fence="false" stretchy="false">}</mo> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{\{1,2\},\{3\}\};\{\{1,3\},\{2\}\};\{\{2,3\},\{1\}\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0abe54c9605021045e52680548da7c640622e5d5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:39.658ex; height:2.843ex;" alt="{\displaystyle \{\{1,2\},\{3\}\};\{\{1,3\},\{2\}\};\{\{2,3\},\{1\}\}}"></span>.</li> <li>Zu den Zahlpartitionen <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 5=3+2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>5</mn> <mo>=</mo> <mn>3</mn> <mo>+</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 5=3+2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ca7529eb70d4179738fa37fa324a4e7df34d152e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.426ex; height:2.343ex;" alt="{\displaystyle 5=3+2}"></span> und <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 5=3+1+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>5</mn> <mo>=</mo> <mn>3</mn> <mo>+</mo> <mn>1</mn> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 5=3+1+1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/06afa895ec9d11eb1b039b48024061139a8022d8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:13.429ex; height:2.343ex;" alt="{\displaystyle 5=3+1+1}"></span> von 5 gehören je <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\binom {5}{3}}=10}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mn>5</mn> <mn>3</mn> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <mo>=</mo> <mn>10</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\binom {5}{3}}=10}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fe3904e5d2155b485f49ee6ffd5f3a6d4d0cb8dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:10.007ex; height:6.176ex;" alt="{\displaystyle {\binom {5}{3}}=10}"></span> Mengenpartitionen, zu den Zahlpartitionen <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 5=5}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>5</mn> <mo>=</mo> <mn>5</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 5=5}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e21c2dc7499b3897681e124151dffd65759e95a0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.423ex; height:2.176ex;" alt="{\displaystyle 5=5}"></span> und <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 5=1+1+1+1+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>5</mn> <mo>=</mo> <mn>1</mn> <mo>+</mo> <mn>1</mn> <mo>+</mo> <mn>1</mn> <mo>+</mo> <mn>1</mn> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 5=1+1+1+1+1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e4d2555e5f0370c9eb1117509c699ce13b37500a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:21.435ex; height:2.343ex;" alt="{\displaystyle 5=1+1+1+1+1}"></span> je genau eine Mengenpartion.</li></ul> <p>Hierbei wird mit Bₙ die n-te <a href="/wiki/Bellsche_Zahl" title="Bellsche Zahl">Bellsche Zahl</a> zum Ausdruck gebracht. </p> <div class="mw-heading mw-heading3"><h3 id="Endliche_abelsche_p-Gruppen_und_abelsche_Gruppen">Endliche abelsche p-Gruppen und abelsche Gruppen</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Partitionsfunktion&veaction=edit&section=23" title="Abschnitt bearbeiten: Endliche abelsche p-Gruppen und abelsche Gruppen" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Partitionsfunktion&action=edit&section=23" title="Quellcode des Abschnitts bearbeiten: Endliche abelsche p-Gruppen und abelsche Gruppen"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Ist <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"></span> eine positive <a href="/wiki/Primzahl" title="Primzahl">Primzahl</a>, dann ist für <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\in \mathbb {N} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>∈<!-- ∈ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\in \mathbb {N} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d059936e77a2d707e9ee0a1d9575a1d693ce5d0b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.913ex; height:2.176ex;" alt="{\displaystyle n\in \mathbb {N} }"></span> jede Gruppe mit der Gruppenordnung <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c6a7a7e74ae90ab94f01e1629177758fb68b423b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:2.477ex; height:2.676ex;" alt="{\displaystyle p^{n}}"></span> eine <a href="/wiki/P-Gruppe" title="P-Gruppe">p-Gruppe</a>. Die Anzahl der (Isomorphieklassen von) <a href="/wiki/Abelsche_Gruppe" title="Abelsche Gruppe">abelschen Gruppen</a> mit <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c6a7a7e74ae90ab94f01e1629177758fb68b423b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:2.477ex; height:2.676ex;" alt="{\displaystyle p^{n}}"></span> Gruppenelementen ist – unabhängig von der Primzahl <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"></span> – gleich dem Wert <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(n)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5e303d2c14cd399b6f52b468c9fd44a542bed422" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.949ex; height:2.843ex;" alt="{\displaystyle P(n)}"></span> der Partitionsfunktion, denn jede solche Gruppe <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}"></span> ist nach dem <a href="/wiki/Hauptsatz_%C3%BCber_endlich_erzeugte_abelsche_Gruppen" title="Hauptsatz über endlich erzeugte abelsche Gruppen">Hauptsatz über endlich erzeugte abelsche Gruppen</a> isomorph zu einem <a href="/wiki/Direktes_Produkt" title="Direktes Produkt">direkten Produkt</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G\cong \mathbb {Z} /p^{k_{1}}\mathbb {Z} \times \mathbb {Z} /p^{k_{2}}\mathbb {Z} \times \cdots \times \mathbb {Z} /p^{k_{r}}\mathbb {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> <mo>≅<!-- ≅ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mo>×<!-- × --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mo>×<!-- × --></mo> <mo>⋯<!-- ⋯ --></mo> <mo>×<!-- × --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msub> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G\cong \mathbb {Z} /p^{k_{1}}\mathbb {Z} \times \mathbb {Z} /p^{k_{2}}\mathbb {Z} \times \cdots \times \mathbb {Z} /p^{k_{r}}\mathbb {Z} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f36bab50dc347fde72b9fffcda8eee668ec576c8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:38.163ex; height:3.176ex;" alt="{\displaystyle G\cong \mathbb {Z} /p^{k_{1}}\mathbb {Z} \times \mathbb {Z} /p^{k_{2}}\mathbb {Z} \times \cdots \times \mathbb {Z} /p^{k_{r}}\mathbb {Z} }"></span> mit <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p^{n}=p^{k_{1}}\cdot p^{k_{2}}\cdot \cdots \cdot p^{k_{r}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>=</mo> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </msup> <mo>⋅<!-- ⋅ --></mo> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> </msup> <mo>⋅<!-- ⋅ --></mo> <mo>⋯<!-- ⋯ --></mo> <mo>⋅<!-- ⋅ --></mo> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msub> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p^{n}=p^{k_{1}}\cdot p^{k_{2}}\cdot \cdots \cdot p^{k_{r}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8853905d92efa6ee3c7139d3d6e57f84b9cf9c62" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:22.54ex; height:3.009ex;" alt="{\displaystyle p^{n}=p^{k_{1}}\cdot p^{k_{2}}\cdot \cdots \cdot p^{k_{r}}}"></span> und also <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n=k_{1}+k_{2}+\cdots +k_{r}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>=</mo> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <mo>⋯<!-- ⋯ --></mo> <mo>+</mo> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n=k_{1}+k_{2}+\cdots +k_{r}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d05ea20144eeb242a0fa0c8eb5f4de6eda367394" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:22.454ex; height:2.509ex;" alt="{\displaystyle n=k_{1}+k_{2}+\cdots +k_{r}}"></span>. Da die Isomorphieklasse nicht von der Reihenfolge der Faktoren im direkten Produkt abhängt, entspricht jede Isomorphieklasse von abelschen Gruppen mit <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c6a7a7e74ae90ab94f01e1629177758fb68b423b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:2.477ex; height:2.676ex;" alt="{\displaystyle p^{n}}"></span> Elementen umkehrbar eindeutig einer Zahlpartition von <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span>. </p><p>Zum Beispiel gibt es bis auf Isomorphie jeweils genau <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(5)=7}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <mn>5</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mn>7</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(5)=7}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e9a438d4ebcd49bf06a1b84e21f71ac4180383f2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.978ex; height:2.843ex;" alt="{\displaystyle P(5)=7}"></span> abelsche Gruppen mit <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 32=2^{5},243=3^{5},3125=5^{5}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>32</mn> <mo>=</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msup> <mo>,</mo> <mn>243</mn> <mo>=</mo> <msup> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msup> <mo>,</mo> <mn>3125</mn> <mo>=</mo> <msup> <mn>5</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 32=2^{5},243=3^{5},3125=5^{5}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f4ea668b05bd8c876b44943e60d5c4a19e0540fd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:28.475ex; height:3.009ex;" alt="{\displaystyle 32=2^{5},243=3^{5},3125=5^{5}}"></span> Elementen. </p><p><b>Anwendungsbeispiele:</b> </p> <ol><li>Wie viele Isomorphietypen von abelschen Gruppen mit genau 70000 Elementen gibt es? Jede solche Gruppe ist, wieder nach dem Hauptsatz ein direktes Produkt ihrer abelschen p-<a href="/wiki/Sylowgruppe" class="mw-redirect" title="Sylowgruppe">Sylowgruppen</a> zu den Primzahlen 2, 5 und 7. Es ist <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 70000=2^{4}\cdot 5^{4}\cdot 7^{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>70000</mn> <mo>=</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>⋅<!-- ⋅ --></mo> <msup> <mn>5</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>⋅<!-- ⋅ --></mo> <msup> <mn>7</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 70000=2^{4}\cdot 5^{4}\cdot 7^{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dd4b47ec1b7a19ef65f8e3c638714edd169cabec" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:18.919ex; height:2.676ex;" alt="{\displaystyle 70000=2^{4}\cdot 5^{4}\cdot 7^{1}}"></span>, also existieren <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(4)\cdot P(4)\cdot P(1)=5\cdot 5\cdot 1=25\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <mn>4</mn> <mo stretchy="false">)</mo> <mo>⋅<!-- ⋅ --></mo> <mi>P</mi> <mo stretchy="false">(</mo> <mn>4</mn> <mo stretchy="false">)</mo> <mo>⋅<!-- ⋅ --></mo> <mi>P</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mn>5</mn> <mo>⋅<!-- ⋅ --></mo> <mn>5</mn> <mo>⋅<!-- ⋅ --></mo> <mn>1</mn> <mo>=</mo> <mn>25</mn> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(4)\cdot P(4)\cdot P(1)=5\cdot 5\cdot 1=25\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/151bb8081231199a36e6a21450d3d79e1766557a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:33.264ex; height:2.843ex;" alt="{\displaystyle P(4)\cdot P(4)\cdot P(1)=5\cdot 5\cdot 1=25\,}"></span> „wesentlich verschiedene“ abelsche Gruppen mit 70000 Elementen.</li> <li>Wie viele Isomorphietypen von abelschen Gruppen mit 7200 Elementen gibt es, die ein Element der Ordnung 180 enthalten? Es ist <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 180=2^{2}\cdot 3^{2}\cdot 5^{1};7200=2^{5}\cdot 3^{2}\cdot 5^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>180</mn> <mo>=</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⋅<!-- ⋅ --></mo> <msup> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⋅<!-- ⋅ --></mo> <msup> <mn>5</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> <mo>;</mo> <mn>7200</mn> <mo>=</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msup> <mo>⋅<!-- ⋅ --></mo> <msup> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⋅<!-- ⋅ --></mo> <msup> <mn>5</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 180=2^{2}\cdot 3^{2}\cdot 5^{1};7200=2^{5}\cdot 3^{2}\cdot 5^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/142c57f97c4b26d5c986c754ee7fb6eaefafb63e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:35.385ex; height:3.009ex;" alt="{\displaystyle 180=2^{2}\cdot 3^{2}\cdot 5^{1};7200=2^{5}\cdot 3^{2}\cdot 5^{2}}"></span>. Von den abelschen 2-Gruppen und 3-Gruppen kommen nur solche in Betracht, die zu einer Partition von 5 bzw. 2 gehören, die einen Summanden größer oder gleich 2 enthält, damit fällt jeweils eine Zahlpartition (Summe von Einsen) weg. Es gibt also <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (P(5)-1)\cdot (P(2)-1)\cdot P(2)=6\cdot 1\cdot 2=12}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>P</mi> <mo stretchy="false">(</mo> <mn>5</mn> <mo stretchy="false">)</mo> <mo>−<!-- − --></mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>⋅<!-- ⋅ --></mo> <mo stretchy="false">(</mo> <mi>P</mi> <mo stretchy="false">(</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo>−<!-- − --></mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>⋅<!-- ⋅ --></mo> <mi>P</mi> <mo stretchy="false">(</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mn>6</mn> <mo>⋅<!-- ⋅ --></mo> <mn>1</mn> <mo>⋅<!-- ⋅ --></mo> <mn>2</mn> <mo>=</mo> <mn>12</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (P(5)-1)\cdot (P(2)-1)\cdot P(2)=6\cdot 1\cdot 2=12}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5573295f913f5403ddd42f9efc0df3f3ad57508e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:44.501ex; height:2.843ex;" alt="{\displaystyle (P(5)-1)\cdot (P(2)-1)\cdot P(2)=6\cdot 1\cdot 2=12}"></span> solche Gruppen.</li> <li>Ist nun <i>zusätzlich</i> zu den Informationen des vorigen Beispiels bekannt, dass kein Element eine <i>größere</i> Ordnung als 180 hat, so kommen nur noch 2 Arten von 2-Sylowgruppen <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (5=2+2+1;5=2+1+1+1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>5</mn> <mo>=</mo> <mn>2</mn> <mo>+</mo> <mn>2</mn> <mo>+</mo> <mn>1</mn> <mo>;</mo> <mn>5</mn> <mo>=</mo> <mn>2</mn> <mo>+</mo> <mn>1</mn> <mo>+</mo> <mn>1</mn> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (5=2+2+1;5=2+1+1+1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1217b6c22b4350041a44d1c0afc8922b387648db" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:33.704ex; height:2.843ex;" alt="{\displaystyle (5=2+2+1;5=2+1+1+1)}"></span> und eine Art 5-Sylowgruppe <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (2=1+1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>2</mn> <mo>=</mo> <mn>1</mn> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (2=1+1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b2ac4ef62c2b5fcb9ce629858a2c24dcfc9c174" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.236ex; height:2.843ex;" alt="{\displaystyle (2=1+1)}"></span> in Betracht und es gibt genau 2 Isomorphietypen von Gruppen mit diesen Eigenschaften.</li></ol> <div class="mw-heading mw-heading4"><h4 id="Anzahlfunktion_von_Isomorphietypen_endlicher_abelscher_Gruppen">Anzahlfunktion von Isomorphietypen endlicher abelscher Gruppen</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Partitionsfunktion&veaction=edit&section=24" title="Abschnitt bearbeiten: Anzahlfunktion von Isomorphietypen endlicher abelscher Gruppen" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Partitionsfunktion&action=edit&section=24" title="Quellcode des Abschnitts bearbeiten: Anzahlfunktion von Isomorphietypen endlicher abelscher Gruppen"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Der Hauptsatz über die endlich erzeugten abelschen Gruppen erlaubt es, die Anzahl <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a(n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a(n)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/943e4e4e0ed21960105be2f6977215ed57e930d9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.434ex; height:2.843ex;" alt="{\displaystyle a(n)}"></span> der Isomorphietypen endlicher abelscher Gruppen mit <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span> Elementen durch die Partitionsfunktion <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}"></span> auszudrücken: </p> <ul><li>Zu jeder natürlichen Zahl <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\in \mathbb {N} \setminus \{0\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>∈<!-- ∈ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> <mo class="MJX-variant">∖<!-- ∖ --></mo> <mo fence="false" stretchy="false">{</mo> <mn>0</mn> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\in \mathbb {N} \setminus \{0\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a0055197414c7cf4bfa79fd73e0c943d14251877" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.596ex; height:2.843ex;" alt="{\displaystyle n\in \mathbb {N} \setminus \{0\}}"></span> mit der <a href="/wiki/Primfaktorzerlegung" title="Primfaktorzerlegung">Primfaktorzerlegung</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n={p_{1}}^{r_{1}}\cdot {p_{2}}^{r_{2}}\cdots {p_{k}}^{r_{k}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>=</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </msup> <mo>⋅<!-- ⋅ --></mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> </msup> <mo>⋯<!-- ⋯ --></mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n={p_{1}}^{r_{1}}\cdot {p_{2}}^{r_{2}}\cdots {p_{k}}^{r_{k}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/09397dc0fd0bc91acca8e6af3d736548b4522dc1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:21.819ex; height:2.676ex;" alt="{\displaystyle n={p_{1}}^{r_{1}}\cdot {p_{2}}^{r_{2}}\cdots {p_{k}}^{r_{k}}}"></span> existieren genau <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a(n)=P(r_{1})\cdot P(r_{2})\cdots P(r_{k})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>P</mi> <mo stretchy="false">(</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>⋅<!-- ⋅ --></mo> <mi>P</mi> <mo stretchy="false">(</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>⋯<!-- ⋯ --></mo> <mi>P</mi> <mo stretchy="false">(</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a(n)=P(r_{1})\cdot P(r_{2})\cdots P(r_{k})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c1f061db2e10ece2d21afdd5ff85bf94ab2b6074" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:29.716ex; height:2.843ex;" alt="{\displaystyle a(n)=P(r_{1})\cdot P(r_{2})\cdots P(r_{k})}"></span> Isomorphietypen von abelschen Gruppen mit <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span> Elementen.</li> <li>Die Folge <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a(n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a(n)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/943e4e4e0ed21960105be2f6977215ed57e930d9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.434ex; height:2.843ex;" alt="{\displaystyle a(n)}"></span> ist Folge <a href="//oeis.org/A000688" class="extiw" title="oeis:A000688">A000688 </a> in <a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>, sie ist eine <i>multiplikative</i> <a href="/wiki/Zahlentheoretische_Funktion" title="Zahlentheoretische Funktion">zahlentheoretische Funktion</a> von <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span> und als solche durch ihre Werte für Primzahlpotenzen vollständig bestimmt.</li> <li>Die der Anzahlfunktion <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a(n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a(n)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/943e4e4e0ed21960105be2f6977215ed57e930d9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.434ex; height:2.843ex;" alt="{\displaystyle a(n)}"></span> zugeordnete (formale) <a href="/wiki/Dirichletreihe" title="Dirichletreihe">Dirichletreihe</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A(s)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A(s)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2f64c6341846d4e1b21bd2d3fdcdc0a31e0b2358" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.643ex; height:2.843ex;" alt="{\displaystyle A(s)}"></span> ist</li></ul> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A(s)=\sum _{n=1}^{\infty }{\frac {a(n)}{n^{s}}},\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munderover> <mo>∑<!-- ∑ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞<!-- ∞ --></mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>a</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msup> </mfrac> </mrow> <mo>,</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A(s)=\sum _{n=1}^{\infty }{\frac {a(n)}{n^{s}}},\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9432e469704522c9098b44e62460295d9fcec681" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:17.787ex; height:6.843ex;" alt="{\displaystyle A(s)=\sum _{n=1}^{\infty }{\frac {a(n)}{n^{s}}},\,}"></span> mit <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s=\sigma +it\in \mathbb {C} ,\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> <mo>=</mo> <mi>σ<!-- σ --></mi> <mo>+</mo> <mi>i</mi> <mi>t</mi> <mo>∈<!-- ∈ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> <mo>,</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s=\sigma +it\in \mathbb {C} ,\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/29d94f8414b840b00b2f42bf7b38f9d769b804dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:15.554ex; height:2.509ex;" alt="{\displaystyle s=\sigma +it\in \mathbb {C} ,\,}"></span> ihr Eulerprodukt lautet <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A(s)=\prod _{p\ {\mathrm {prim} }}\sum _{r=0}^{\infty }{\frac {P(r)}{p^{rs}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munder> <mo>∏<!-- ∏ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">p</mi> <mi mathvariant="normal">r</mi> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">m</mi> </mrow> </mrow> </mrow> </munder> <munderover> <mo>∑<!-- ∑ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞<!-- ∞ --></mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>P</mi> <mo stretchy="false">(</mo> <mi>r</mi> <mo stretchy="false">)</mo> </mrow> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> <mi>s</mi> </mrow> </msup> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A(s)=\prod _{p\ {\mathrm {prim} }}\sum _{r=0}^{\infty }{\frac {P(r)}{p^{rs}}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/38947381ea6650a05da19c246cd89a6de11da203" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:22.749ex; height:7.176ex;" alt="{\displaystyle A(s)=\prod _{p\ {\mathrm {prim} }}\sum _{r=0}^{\infty }{\frac {P(r)}{p^{rs}}}.}"></span></dd></dl> <ul><li>Die Anzahlfunktion <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a(n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a(n)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/943e4e4e0ed21960105be2f6977215ed57e930d9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.434ex; height:2.843ex;" alt="{\displaystyle a(n)}"></span> gibt für <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\geq 2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>≥<!-- ≥ --></mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\geq 2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e6bf67f9d06ca3af619657f8d20ee1322da77174" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.656ex; height:2.343ex;" alt="{\displaystyle n\geq 2}"></span> zugleich die Anzahl der durch die <a href="/wiki/Teilbarkeit" title="Teilbarkeit">Teilbarkeitsrelation</a> geordneten Ketten <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (t_{1},t_{2},\ldots ,t_{r})\in \left\lbrace \left(\mathbb {N} \setminus \{0,1\}\right)^{r}:\;t_{1}|t_{2}|\cdots |t_{r}\right\rbrace }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>…<!-- … --></mo> <mo>,</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>∈<!-- ∈ --></mo> <mrow> <mo>{</mo> <mrow> <msup> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> <mo class="MJX-variant">∖<!-- ∖ --></mo> <mo fence="false" stretchy="false">{</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo fence="false" stretchy="false">}</mo> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msup> <mo>:</mo> <mspace width="thickmathspace" /> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>⋯<!-- ⋯ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msub> </mrow> <mo>}</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (t_{1},t_{2},\ldots ,t_{r})\in \left\lbrace \left(\mathbb {N} \setminus \{0,1\}\right)^{r}:\;t_{1}|t_{2}|\cdots |t_{r}\right\rbrace }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/729fb8a13e81f31d52fbd0b736e247b3de4a2a29" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:44.749ex; height:3.009ex;" alt="{\displaystyle (t_{1},t_{2},\ldots ,t_{r})\in \left\lbrace \left(\mathbb {N} \setminus \{0,1\}\right)^{r}:\;t_{1}|t_{2}|\cdots |t_{r}\right\rbrace }"></span> an, deren Produkt gleich <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span> ist <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (t_{1}\cdot t_{2}\cdot \cdots t_{r}=n).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>⋅<!-- ⋅ --></mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>⋅<!-- ⋅ --></mo> <mo>⋯<!-- ⋯ --></mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msub> <mo>=</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (t_{1}\cdot t_{2}\cdot \cdots t_{r}=n).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6835c31f7b662cf06a3c1d4c339db5ba0399c76f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.019ex; height:2.843ex;" alt="{\displaystyle (t_{1}\cdot t_{2}\cdot \cdots t_{r}=n).}"></span></li></ul> <div class="mw-heading mw-heading2"><h2 id="Strikte_Partitionsfunktion">Strikte Partitionsfunktion</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Partitionsfunktion&veaction=edit&section=25" title="Abschnitt bearbeiten: Strikte Partitionsfunktion" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Partitionsfunktion&action=edit&section=25" title="Quellcode des Abschnitts bearbeiten: Strikte Partitionsfunktion"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Definition_und_Eigenschaften_der_strikten_Partitionen">Definition und Eigenschaften der strikten Partitionen</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Partitionsfunktion&veaction=edit&section=26" title="Abschnitt bearbeiten: Definition und Eigenschaften der strikten Partitionen" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Partitionsfunktion&action=edit&section=26" title="Quellcode des Abschnitts bearbeiten: Definition und Eigenschaften der strikten Partitionen"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Wenn jeder Summand nur einmal<sup id="cite_ref-14" class="reference"><a href="#cite_note-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> in der Partitionssumme vorkommen darf und somit kein Summand wiederholt in der Partitionssumme auftaucht, dann liegen die sogenannten strikten Partitionen vor. Die strikte Partitionsfolge Q(n) erfüllt somit für alle n ∈ ℕ₀ das Kriterium Q(n) ≤ P(n). Die gleiche Folge<sup id="cite_ref-15" class="reference"><a href="#cite_note-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> ergibt sich, wenn in der Partitionssumme nur ungerade Summanden<sup id="cite_ref-16" class="reference"><a href="#cite_note-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup> auftauchen dürfen, aber diese auch mehrfach vorkommen dürfen. </p> <div class="mw-heading mw-heading3"><h3 id="Beispielwerte_der_strikten_Partitionszahlen">Beispielwerte der strikten Partitionszahlen</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Partitionsfunktion&veaction=edit&section=27" title="Abschnitt bearbeiten: Beispielwerte der strikten Partitionszahlen" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Partitionsfunktion&action=edit&section=27" title="Quellcode des Abschnitts bearbeiten: Beispielwerte der strikten Partitionszahlen"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Darstellungen der Partitionen: </p> <table class="wikitable"> <caption>Beispielwerte von Q(n) und zugehörige Zahlpartitionen </caption> <tbody><tr> <th>n</th> <th>Q(n)</th> <th>Zahlpartitionen ohne wiederholte Summanden </th> <th>Zahlpartitionen mit nur ungeraden Summanden </th></tr> <tr> <td>0</td> <td>1</td> <td>() leere Partition/<a href="/wiki/Leere_Summe" title="Leere Summe">leere Summe</a> </td> <td>() leere Partition/<a href="/wiki/Leere_Summe" title="Leere Summe">leere Summe</a> </td></tr> <tr> <td>1</td> <td>1</td> <td>(1) </td> <td>(1) </td></tr> <tr> <td>2</td> <td>1</td> <td>(2) </td> <td>(1+1) </td></tr> <tr> <td>3</td> <td>2</td> <td>(1+2), (3) </td> <td>(1+1+1), (3) </td></tr> <tr> <td>4</td> <td>2</td> <td>(1+3), (4) </td> <td>(1+1+1+1), (1+3) </td></tr> <tr> <td>5</td> <td>3</td> <td>(2+3), (1+4), (5) </td> <td>(1+1+1+1+1), (1+1+3), (5) </td></tr> <tr> <td>6</td> <td>4</td> <td>(1+2+3), (2+4), (1+5), (6) </td> <td>(1+1+1+1+1+1), (1+1+1+3), (3+3), (1+5) </td></tr> <tr> <td>7 </td> <td>5 </td> <td>(1+2+4), (3+4), (2+5), (1+6), (7) </td> <td>(1+1+1+1+1+1+1), (1+1+1+1+3), (1+3+3), (1+1+5), (7) </td></tr> <tr> <td>8 </td> <td>6 </td> <td>(1+3+4), (1+2+5), (3+5), (2+6), (1+7), (8) </td> <td>(1+1+1+1+1+1+1+1), (1+1+1+1+1+3), (1+1+3+3), (1+1+1+5), (3+5), (1+7) </td></tr> <tr> <td>9 </td> <td>8 </td> <td>(2+3+4), (1+3+5), (4+5), (1+2+6), (3+6), (2+7), (1+8), (9) </td> <td>...... </td></tr> <tr> <td>10 </td> <td>10 </td> <td>(1+2+3+4), (2+3+5), (1+4+5), (1+3+6), (4+6), (1+2+7), (3+7), (2+8), (1+9), (10) </td> <td>...... </td></tr></tbody></table> <div class="mw-heading mw-heading3"><h3 id="Maclaurinsche_Reihe_der_strikten_Partitionszahlen">Maclaurinsche Reihe der strikten Partitionszahlen</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Partitionsfunktion&veaction=edit&section=28" title="Abschnitt bearbeiten: Maclaurinsche Reihe der strikten Partitionszahlen" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Partitionsfunktion&action=edit&section=28" title="Quellcode des Abschnitts bearbeiten: Maclaurinsche Reihe der strikten Partitionszahlen"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Die zugehörige erzeugende Funktion basierend auf der <a href="/wiki/Maclaurinsche_Reihe" title="Maclaurinsche Reihe">Maclaurinschen Reihe</a> mit den Zahlen Q(n) als Koeffizienten vor xⁿ lautet so: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{n=0}^{\infty }Q(n)x^{n}=\prod _{m=1}^{\infty }(1+x^{m})={\tfrac {1}{2}}(-1;x)_{\infty }=(x;x^{2})_{\infty }^{-1}={\sqrt[{6}]{\psi _{R}(x^{2})\vartheta _{00}(x)\vartheta _{01}(x)^{-2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∑<!-- ∑ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞<!-- ∞ --></mi> </mrow> </munderover> <mi>Q</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>=</mo> <munderover> <mo>∏<!-- ∏ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞<!-- ∞ --></mi> </mrow> </munderover> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mo stretchy="false">(</mo> <mo>−<!-- − --></mo> <mn>1</mn> <mo>;</mo> <mi>x</mi> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞<!-- ∞ --></mi> </mrow> </msub> <mo>=</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>;</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msubsup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞<!-- ∞ --></mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−<!-- − --></mo> <mn>1</mn> </mrow> </msubsup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mroot> <mrow> <msub> <mi>ψ<!-- ψ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>R</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> <msub> <mi>ϑ<!-- ϑ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>00</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <msub> <mi>ϑ<!-- ϑ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>01</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−<!-- − --></mo> <mn>2</mn> </mrow> </msup> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </mroot> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{n=0}^{\infty }Q(n)x^{n}=\prod _{m=1}^{\infty }(1+x^{m})={\tfrac {1}{2}}(-1;x)_{\infty }=(x;x^{2})_{\infty }^{-1}={\sqrt[{6}]{\psi _{R}(x^{2})\vartheta _{00}(x)\vartheta _{01}(x)^{-2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/457b5359f5c313f66907f8861c486c607913f1c1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:80.274ex; height:6.843ex;" alt="{\displaystyle \sum _{n=0}^{\infty }Q(n)x^{n}=\prod _{m=1}^{\infty }(1+x^{m})={\tfrac {1}{2}}(-1;x)_{\infty }=(x;x^{2})_{\infty }^{-1}={\sqrt[{6}]{\psi _{R}(x^{2})\vartheta _{00}(x)\vartheta _{01}(x)^{-2}}}}"></span></dd></dl> <p>Man erhält folgende erste Summanden: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x;x^{2})_{\infty }^{-1}={\frac {1}{\prod _{k=1}^{\infty }(1-x^{2k-1})}}=1+1x+1x^{2}+2x^{3}+2x^{4}+3x^{5}+4x^{6}+5x^{7}+6x^{8}+8x^{9}+10x^{10}...}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>x</mi> <mo>;</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msubsup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞<!-- ∞ --></mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−<!-- − --></mo> <mn>1</mn> </mrow> </msubsup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <munderover> <mo>∏<!-- ∏ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞<!-- ∞ --></mi> </mrow> </munderover> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−<!-- − --></mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>k</mi> <mo>−<!-- − --></mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>=</mo> <mn>1</mn> <mo>+</mo> <mn>1</mn> <mi>x</mi> <mo>+</mo> <mn>1</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>2</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>+</mo> <mn>2</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>+</mo> <mn>3</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msup> <mo>+</mo> <mn>4</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msup> <mo>+</mo> <mn>5</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>7</mn> </mrow> </msup> <mo>+</mo> <mn>6</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>8</mn> </mrow> </msup> <mo>+</mo> <mn>8</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>9</mn> </mrow> </msup> <mo>+</mo> <mn>10</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>10</mn> </mrow> </msup> <mo>.</mo> <mo>.</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x;x^{2})_{\infty }^{-1}={\frac {1}{\prod _{k=1}^{\infty }(1-x^{2k-1})}}=1+1x+1x^{2}+2x^{3}+2x^{4}+3x^{5}+4x^{6}+5x^{7}+6x^{8}+8x^{9}+10x^{10}...}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c9a0974d68ffcf38ff9f7d466dbf3456c604064e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:101.134ex; height:6.176ex;" alt="{\displaystyle (x;x^{2})_{\infty }^{-1}={\frac {1}{\prod _{k=1}^{\infty }(1-x^{2k-1})}}=1+1x+1x^{2}+2x^{3}+2x^{4}+3x^{5}+4x^{6}+5x^{7}+6x^{8}+8x^{9}+10x^{10}...}"></span></dd></dl> <p>Wichtige Rechenhinweise über die <a href="/wiki/Thetafunktion" class="mw-redirect" title="Thetafunktion">Thetafunktionen</a> und die <a href="/wiki/Ramanujansche_Psifunktion" title="Ramanujansche Psifunktion">Ramanujansche Psifunktion</a>: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x;x)_{\infty }(x;x^{2})_{\infty }=\vartheta _{01}(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>x</mi> <mo>;</mo> <mi>x</mi> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞<!-- ∞ --></mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo>;</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞<!-- ∞ --></mi> </mrow> </msub> <mo>=</mo> <msub> <mi>ϑ<!-- ϑ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>01</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x;x)_{\infty }(x;x^{2})_{\infty }=\vartheta _{01}(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b9b001ac449f700fbe9f55d7b550d899489eddc6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.298ex; height:3.176ex;" alt="{\displaystyle (x;x)_{\infty }(x;x^{2})_{\infty }=\vartheta _{01}(x)}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x;x)_{\infty }^{2}=(x;x^{2})_{\infty }^{4}\psi _{R}(x^{2})\vartheta _{00}(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>x</mi> <mo>;</mo> <mi>x</mi> <msubsup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞<!-- ∞ --></mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>=</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>;</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msubsup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞<!-- ∞ --></mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msubsup> <msub> <mi>ψ<!-- ψ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>R</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> <msub> <mi>ϑ<!-- ϑ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>00</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x;x)_{\infty }^{2}=(x;x^{2})_{\infty }^{4}\psi _{R}(x^{2})\vartheta _{00}(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eeb2ec5f714d208e3ab5062c4be6efea993478d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:32.484ex; height:3.176ex;" alt="{\displaystyle (x;x)_{\infty }^{2}=(x;x^{2})_{\infty }^{4}\psi _{R}(x^{2})\vartheta _{00}(x)}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 16\,x\,\psi _{R}(x^{2})^{4}=\vartheta _{00}(x)^{4}-\vartheta _{01}(x)^{4}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>16</mn> <mspace width="thinmathspace" /> <mi>x</mi> <mspace width="thinmathspace" /> <msub> <mi>ψ<!-- ψ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>R</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>=</mo> <msub> <mi>ϑ<!-- ϑ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>00</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>−<!-- − --></mo> <msub> <mi>ϑ<!-- ϑ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>01</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 16\,x\,\psi _{R}(x^{2})^{4}=\vartheta _{00}(x)^{4}-\vartheta _{01}(x)^{4}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ce9e954e192d345777ea033a62bb5a094bf4271c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:33.494ex; height:3.176ex;" alt="{\displaystyle 16\,x\,\psi _{R}(x^{2})^{4}=\vartheta _{00}(x)^{4}-\vartheta _{01}(x)^{4}}"></span></dd></dl> <p>Alle drei hier abgebildeten Formeln gelten für den Definitionsbereich −1 < x < 1. </p> <div class="mw-heading mw-heading3"><h3 id="Identitäten_über_die_strikte_Partitionsfunktion"><span id="Identit.C3.A4ten_.C3.BCber_die_strikte_Partitionsfunktion"></span>Identitäten über die strikte Partitionsfunktion</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Partitionsfunktion&veaction=edit&section=29" title="Abschnitt bearbeiten: Identitäten über die strikte Partitionsfunktion" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Partitionsfunktion&action=edit&section=29" title="Quellcode des Abschnitts bearbeiten: Identitäten über die strikte Partitionsfunktion"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Über die Pochhammer-Produkte gilt weiter folgende Identität: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x;x)_{\infty }^{-1}=(x^{2};x^{2})_{\infty }^{-1}(x;x^{2})_{\infty }^{-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>x</mi> <mo>;</mo> <mi>x</mi> <msubsup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞<!-- ∞ --></mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−<!-- − --></mo> <mn>1</mn> </mrow> </msubsup> <mo>=</mo> <mo stretchy="false">(</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>;</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msubsup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞<!-- ∞ --></mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−<!-- − --></mo> <mn>1</mn> </mrow> </msubsup> <mo stretchy="false">(</mo> <mi>x</mi> <mo>;</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msubsup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞<!-- ∞ --></mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−<!-- − --></mo> <mn>1</mn> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x;x)_{\infty }^{-1}=(x^{2};x^{2})_{\infty }^{-1}(x;x^{2})_{\infty }^{-1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/427833fe7d689a4f6fa9a170352a85af860e7742" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:29.767ex; height:3.176ex;" alt="{\displaystyle (x;x)_{\infty }^{-1}=(x^{2};x^{2})_{\infty }^{-1}(x;x^{2})_{\infty }^{-1}}"></span></dd></dl> <p>Daraus folgt diese Formel: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\biggl [}\sum _{n=0}^{\infty }P(n)x^{n}{\biggr ]}={\biggl [}\sum _{n=0}^{\infty }P(n)x^{2n}{\biggr ]}{\biggl [}\sum _{n=0}^{\infty }Q(n)x^{n}{\biggr ]}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">[</mo> </mrow> </mrow> <munderover> <mo>∑<!-- ∑ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞<!-- ∞ --></mi> </mrow> </munderover> <mi>P</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">]</mo> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">[</mo> </mrow> </mrow> <munderover> <mo>∑<!-- ∑ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞<!-- ∞ --></mi> </mrow> </munderover> <mi>P</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>n</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">]</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">[</mo> </mrow> </mrow> <munderover> <mo>∑<!-- ∑ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞<!-- ∞ --></mi> </mrow> </munderover> <mi>Q</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\biggl [}\sum _{n=0}^{\infty }P(n)x^{n}{\biggr ]}={\biggl [}\sum _{n=0}^{\infty }P(n)x^{2n}{\biggr ]}{\biggl [}\sum _{n=0}^{\infty }Q(n)x^{n}{\biggr ]}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f2b65a7beb8a2cb56fe65c49fdf1a535fc5cd306" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:46.259ex; height:6.843ex;" alt="{\displaystyle {\biggl [}\sum _{n=0}^{\infty }P(n)x^{n}{\biggr ]}={\biggl [}\sum _{n=0}^{\infty }P(n)x^{2n}{\biggr ]}{\biggl [}\sum _{n=0}^{\infty }Q(n)x^{n}{\biggr ]}}"></span></dd></dl> <p>Und deswegen gelten auch jene zwei Formeln für die Synthese der Zahlenfolge P(n): </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(2n)=\sum _{k=0}^{n}P(n-k)Q(2k)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <mn>2</mn> <mi>n</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munderover> <mo>∑<!-- ∑ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mi>P</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−<!-- − --></mo> <mi>k</mi> <mo stretchy="false">)</mo> <mi>Q</mi> <mo stretchy="false">(</mo> <mn>2</mn> <mi>k</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(2n)=\sum _{k=0}^{n}P(n-k)Q(2k)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c99280734ad1464569459c103153af0c46e54b51" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:27.975ex; height:7.009ex;" alt="{\displaystyle P(2n)=\sum _{k=0}^{n}P(n-k)Q(2k)}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(2n+1)=\sum _{k=0}^{n}P(n-k)Q(2k+1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <mn>2</mn> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>=</mo> <munderover> <mo>∑<!-- ∑ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mi>P</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−<!-- − --></mo> <mi>k</mi> <mo stretchy="false">)</mo> <mi>Q</mi> <mo stretchy="false">(</mo> <mn>2</mn> <mi>k</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(2n+1)=\sum _{k=0}^{n}P(n-k)Q(2k+1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b9d7d0f1f129bb9727a785c1d757a92b78cf469f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:35.981ex; height:7.009ex;" alt="{\displaystyle P(2n+1)=\sum _{k=0}^{n}P(n-k)Q(2k+1)}"></span></dd></dl> <p>Im nun Folgenden werden zwei Beispiele akkurat ausgeführt: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(8)=\sum _{k=0}^{4}P(4-k)Q(2k)=}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <mn>8</mn> <mo stretchy="false">)</mo> <mo>=</mo> <munderover> <mo>∑<!-- ∑ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </munderover> <mi>P</mi> <mo stretchy="false">(</mo> <mn>4</mn> <mo>−<!-- − --></mo> <mi>k</mi> <mo stretchy="false">)</mo> <mi>Q</mi> <mo stretchy="false">(</mo> <mn>2</mn> <mi>k</mi> <mo stretchy="false">)</mo> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(8)=\sum _{k=0}^{4}P(4-k)Q(2k)=}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/91741e89655ec852e963eb1c43e589e014206247" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:28.801ex; height:7.509ex;" alt="{\displaystyle P(8)=\sum _{k=0}^{4}P(4-k)Q(2k)=}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle =P(4)Q(0)+P(3)Q(2)+P(2)Q(4)+P(1)Q(6)+P(0)Q(8)=}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>=</mo> <mi>P</mi> <mo stretchy="false">(</mo> <mn>4</mn> <mo stretchy="false">)</mo> <mi>Q</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>+</mo> <mi>P</mi> <mo stretchy="false">(</mo> <mn>3</mn> <mo stretchy="false">)</mo> <mi>Q</mi> <mo stretchy="false">(</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo>+</mo> <mi>P</mi> <mo stretchy="false">(</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mi>Q</mi> <mo stretchy="false">(</mo> <mn>4</mn> <mo stretchy="false">)</mo> <mo>+</mo> <mi>P</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mi>Q</mi> <mo stretchy="false">(</mo> <mn>6</mn> <mo stretchy="false">)</mo> <mo>+</mo> <mi>P</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mi>Q</mi> <mo stretchy="false">(</mo> <mn>8</mn> <mo stretchy="false">)</mo> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle =P(4)Q(0)+P(3)Q(2)+P(2)Q(4)+P(1)Q(6)+P(0)Q(8)=}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7fe9ad49ac6ed03abb8f7ce2a6b8e9426f12ccfc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:63.904ex; height:2.843ex;" alt="{\displaystyle =P(4)Q(0)+P(3)Q(2)+P(2)Q(4)+P(1)Q(6)+P(0)Q(8)=}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle =5\times 1+3\times 1+2\times 2+1\times 4+1\times 6=22}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>=</mo> <mn>5</mn> <mo>×<!-- × --></mo> <mn>1</mn> <mo>+</mo> <mn>3</mn> <mo>×<!-- × --></mo> <mn>1</mn> <mo>+</mo> <mn>2</mn> <mo>×<!-- × --></mo> <mn>2</mn> <mo>+</mo> <mn>1</mn> <mo>×<!-- × --></mo> <mn>4</mn> <mo>+</mo> <mn>1</mn> <mo>×<!-- × --></mo> <mn>6</mn> <mo>=</mo> <mn>22</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle =5\times 1+3\times 1+2\times 2+1\times 4+1\times 6=22}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e2b3b756dd31de1c5af46043a54463fe89e28992" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:45.065ex; height:2.343ex;" alt="{\displaystyle =5\times 1+3\times 1+2\times 2+1\times 4+1\times 6=22}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(9)=\sum _{k=0}^{4}P(4-k)Q(2k+1)=}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <mn>9</mn> <mo stretchy="false">)</mo> <mo>=</mo> <munderover> <mo>∑<!-- ∑ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </munderover> <mi>P</mi> <mo stretchy="false">(</mo> <mn>4</mn> <mo>−<!-- − --></mo> <mi>k</mi> <mo stretchy="false">)</mo> <mi>Q</mi> <mo stretchy="false">(</mo> <mn>2</mn> <mi>k</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(9)=\sum _{k=0}^{4}P(4-k)Q(2k+1)=}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4267b000f1ecbd3402a03390c13ea10583cbec39" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:32.804ex; height:7.509ex;" alt="{\displaystyle P(9)=\sum _{k=0}^{4}P(4-k)Q(2k+1)=}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle =P(4)Q(1)+P(3)Q(3)+P(2)Q(5)+P(1)Q(7)+P(0)Q(9)=}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>=</mo> <mi>P</mi> <mo stretchy="false">(</mo> <mn>4</mn> <mo stretchy="false">)</mo> <mi>Q</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>+</mo> <mi>P</mi> <mo stretchy="false">(</mo> <mn>3</mn> <mo stretchy="false">)</mo> <mi>Q</mi> <mo stretchy="false">(</mo> <mn>3</mn> <mo stretchy="false">)</mo> <mo>+</mo> <mi>P</mi> <mo stretchy="false">(</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mi>Q</mi> <mo stretchy="false">(</mo> <mn>5</mn> <mo stretchy="false">)</mo> <mo>+</mo> <mi>P</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mi>Q</mi> <mo stretchy="false">(</mo> <mn>7</mn> <mo stretchy="false">)</mo> <mo>+</mo> <mi>P</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mi>Q</mi> <mo stretchy="false">(</mo> <mn>9</mn> <mo stretchy="false">)</mo> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle =P(4)Q(1)+P(3)Q(3)+P(2)Q(5)+P(1)Q(7)+P(0)Q(9)=}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/91bfca45a03bf565468dca38fffec77f706d4ae7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:63.904ex; height:2.843ex;" alt="{\displaystyle =P(4)Q(1)+P(3)Q(3)+P(2)Q(5)+P(1)Q(7)+P(0)Q(9)=}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle =5\times 1+3\times 2+2\times 3+1\times 5+1\times 8=30}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>=</mo> <mn>5</mn> <mo>×<!-- × --></mo> <mn>1</mn> <mo>+</mo> <mn>3</mn> <mo>×<!-- × --></mo> <mn>2</mn> <mo>+</mo> <mn>2</mn> <mo>×<!-- × --></mo> <mn>3</mn> <mo>+</mo> <mn>1</mn> <mo>×<!-- × --></mo> <mn>5</mn> <mo>+</mo> <mn>1</mn> <mo>×<!-- × --></mo> <mn>8</mn> <mo>=</mo> <mn>30</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle =5\times 1+3\times 2+2\times 3+1\times 5+1\times 8=30}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/162d956736c15b9799e2f8950d77949238518732" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:45.065ex; height:2.343ex;" alt="{\displaystyle =5\times 1+3\times 2+2\times 3+1\times 5+1\times 8=30}"></span></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Oberpartitionsfunktion">Oberpartitionsfunktion</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Partitionsfunktion&veaction=edit&section=30" title="Abschnitt bearbeiten: Oberpartitionsfunktion" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Partitionsfunktion&action=edit&section=30" title="Quellcode des Abschnitts bearbeiten: Oberpartitionsfunktion"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Definition_der_Oberpartitionen">Definition der Oberpartitionen</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Partitionsfunktion&veaction=edit&section=31" title="Abschnitt bearbeiten: Definition der Oberpartitionen" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Partitionsfunktion&action=edit&section=31" title="Quellcode des Abschnitts bearbeiten: Definition der Oberpartitionen"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Wenn zu einer gegebenen Zahl k alle Partitionen so aufgestellt werden, dass die Summandengröße niemals steigt, und bei jeder so beschaffenen Partition all diejenigen Summanden markiert werden dürfen, welche keinen gleich großen Summanden links von sich haben, dann wird die sich dadurch ergebende Anzahl der markierten Partitionen in Abhängigkeit von k durch die Oberpartitionsfunktion <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overline {P}}(k)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>P</mi> <mo accent="false">¯<!-- ¯ --></mo> </mover> </mrow> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {P}}(k)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/35b56dad80aca6fc299beb6e95ceb9bef67f7413" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.046ex; height:3.509ex;" alt="{\displaystyle {\overline {P}}(k)}"></span> beschrieben. Diese Funktion kann auch Überpartitionsfunktion genannt werden und wird im englischen Sprachraum <i>overpartition function</i> genannt. </p> <div class="mw-heading mw-heading3"><h3 id="Beispielwerte_der_Oberpartitionszahlen">Beispielwerte der Oberpartitionszahlen</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Partitionsfunktion&veaction=edit&section=32" title="Abschnitt bearbeiten: Beispielwerte der Oberpartitionszahlen" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Partitionsfunktion&action=edit&section=32" title="Quellcode des Abschnitts bearbeiten: Beispielwerte der Oberpartitionszahlen"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Im nun Folgenden werden die ersten Oberpartitionsfunktionswerte mit ihrer beschreibenden Darstellung tabellarisch aufgelistet: </p> <table class="wikitable"> <tbody><tr> <th>n</th> <th><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overline {P}}(n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>P</mi> <mo accent="false">¯<!-- ¯ --></mo> </mover> </mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {P}}(n)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8873a539ad8a7faddae443059099f07b192c9995" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.229ex; height:3.509ex;" alt="{\displaystyle {\overline {P}}(n)}"></span></th> <th>Markierte Zahlpartitionen mit Markierungen nicht wiederholter Summanden </th></tr> <tr> <td>0</td> <td>1</td> <td>() leere Partition </td></tr> <tr> <td>1</td> <td>2</td> <td>(1), (<b>1</b>) </td></tr> <tr> <td>2</td> <td>4</td> <td>(2), (<b>2</b>), (1+1), (<b>1</b>+1) </td></tr> <tr> <td>3</td> <td>8</td> <td>(3), (<b>3</b>), (2+1), (<b>2</b>+1), (2+<b>1</b>), (<b>2</b>+<b>1</b>), (1+1+1), (<b>1</b>+1+1) </td></tr> <tr> <td>4</td> <td>14</td> <td>(4), (<b>4</b>), (3+1), (<b>3</b>+1), (3+<b>1</b>), (<b>3</b>+<b>1</b>), (2+2), (<b>2</b>+2), (2+1+1), (<b>2</b>+1+1), (2+<b>1</b>+1), (<b>2</b>+<b>1</b>+1), (1+1+1+1), (<b>1</b>+1+1+1) </td></tr> <tr> <td>5</td> <td>24</td> <td>(5), (<b>5</b>), (4+1), (<b>4</b>+1), (4+<b>1</b>), (<b>4</b>+<b>1</b>), (3+2), (<b>3</b>+2), (3+<b>2</b>), (<b>3</b>+<b>2</b>), (3+1+1), (<b>3</b>+1+1), (3+<b>1</b>+1), (<b>3</b>+<b>1</b>+1), (2+2+1), (<b>2</b>+2+1), (2+2+<b>1</b>), (<b>2</b>+2+<b>1</b>), <p>(2+1+1+1), (<b>2</b>+1+1+1), (2+<b>1</b>+1+1), (<b>2</b>+<b>1</b>+1+1), (1+1+1+1+1), (<b>1</b>+1+1+1+1) </p> </td></tr></tbody></table> <div class="mw-heading mw-heading3"><h3 id="Maclaurinsche_Reihe_der_Oberpartitionszahlen">Maclaurinsche Reihe der Oberpartitionszahlen</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Partitionsfunktion&veaction=edit&section=33" title="Abschnitt bearbeiten: Maclaurinsche Reihe der Oberpartitionszahlen" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Partitionsfunktion&action=edit&section=33" title="Quellcode des Abschnitts bearbeiten: Maclaurinsche Reihe der Oberpartitionszahlen"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Die erzeugende Funktion der Oberpartitionszahlen <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overline {P}}(n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>P</mi> <mo accent="false">¯<!-- ¯ --></mo> </mover> </mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {P}}(n)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8873a539ad8a7faddae443059099f07b192c9995" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.229ex; height:3.509ex;" alt="{\displaystyle {\overline {P}}(n)}"></span> als Koeffizienten vor <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/150d38e238991bc4d0689ffc9d2a852547d2658d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.548ex; height:2.343ex;" alt="{\displaystyle x^{n}}"></span> ist der Kehrwert der Thetafunktion ϑ₀₁(x): </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{n=0}^{\infty }{\overline {P}}(n)x^{n}=\prod _{k=1}^{\infty }{\frac {1+x^{k}}{1-x^{k}}}=\prod _{k=1}^{\infty }{\frac {1-x^{2k}}{(1-x^{k})^{2}}}={\frac {(x^{2};x^{2})_{\infty }}{(x;x)_{\infty }^{2}}}={\frac {1}{(x;x)_{\infty }(x;x^{2})_{\infty }}}={\frac {1}{\vartheta _{01}(x)}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∑<!-- ∑ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞<!-- ∞ --></mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>P</mi> <mo accent="false">¯<!-- ¯ --></mo> </mover> </mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>=</mo> <munderover> <mo>∏<!-- ∏ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞<!-- ∞ --></mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>1</mn> <mo>+</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> </mrow> <mrow> <mn>1</mn> <mo>−<!-- − --></mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>=</mo> <munderover> <mo>∏<!-- ∏ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞<!-- ∞ --></mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>1</mn> <mo>−<!-- − --></mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>k</mi> </mrow> </msup> </mrow> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−<!-- − --></mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>;</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞<!-- ∞ --></mi> </mrow> </msub> </mrow> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>;</mo> <mi>x</mi> <msubsup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞<!-- ∞ --></mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>;</mo> <mi>x</mi> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞<!-- ∞ --></mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo>;</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞<!-- ∞ --></mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <msub> <mi>ϑ<!-- ϑ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>01</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{n=0}^{\infty }{\overline {P}}(n)x^{n}=\prod _{k=1}^{\infty }{\frac {1+x^{k}}{1-x^{k}}}=\prod _{k=1}^{\infty }{\frac {1-x^{2k}}{(1-x^{k})^{2}}}={\frac {(x^{2};x^{2})_{\infty }}{(x;x)_{\infty }^{2}}}={\frac {1}{(x;x)_{\infty }(x;x^{2})_{\infty }}}={\frac {1}{\vartheta _{01}(x)}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/95ab8158142e0a5d0186039291de8cb73f0d22f9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:85.297ex; height:7.009ex;" alt="{\displaystyle \sum _{n=0}^{\infty }{\overline {P}}(n)x^{n}=\prod _{k=1}^{\infty }{\frac {1+x^{k}}{1-x^{k}}}=\prod _{k=1}^{\infty }{\frac {1-x^{2k}}{(1-x^{k})^{2}}}={\frac {(x^{2};x^{2})_{\infty }}{(x;x)_{\infty }^{2}}}={\frac {1}{(x;x)_{\infty }(x;x^{2})_{\infty }}}={\frac {1}{\vartheta _{01}(x)}}}"></span></dd></dl> <p>Man erhält folgende erste Summanden: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{\vartheta _{01}(x)}}=1+2x+4x^{2}+8x^{3}+14x^{4}+24x^{5}+...}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <msub> <mi>ϑ<!-- ϑ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>01</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>=</mo> <mn>1</mn> <mo>+</mo> <mn>2</mn> <mi>x</mi> <mo>+</mo> <mn>4</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>8</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>+</mo> <mn>14</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>+</mo> <mn>24</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msup> <mo>+</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{\vartheta _{01}(x)}}=1+2x+4x^{2}+8x^{3}+14x^{4}+24x^{5}+...}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1546f9410816120f8fb211570f0da7a2dca73210" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:49.213ex; height:6.009ex;" alt="{\displaystyle {\frac {1}{\vartheta _{01}(x)}}=1+2x+4x^{2}+8x^{3}+14x^{4}+24x^{5}+...}"></span></dd></dl> <p>Im Vergleich hierzu gilt für die Thetafunktion ϑ₀₁(x) selbst diese Formel: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \vartheta _{01}(x)=1-{\biggl \{}\sum _{k=1}^{\infty }2{\bigl [}x^{(2k-1)^{2}}-x^{(2k)^{2}}{\bigr ]}{\biggr \}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ϑ<!-- ϑ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>01</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1</mn> <mo>−<!-- − --></mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">{</mo> </mrow> </mrow> <munderover> <mo>∑<!-- ∑ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞<!-- ∞ --></mi> </mrow> </munderover> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">[</mo> </mrow> </mrow> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>2</mn> <mi>k</mi> <mo>−<!-- − --></mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </msup> <mo>−<!-- − --></mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>2</mn> <mi>k</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">]</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">}</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \vartheta _{01}(x)=1-{\biggl \{}\sum _{k=1}^{\infty }2{\bigl [}x^{(2k-1)^{2}}-x^{(2k)^{2}}{\bigr ]}{\biggr \}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a0e5a633a778861b2bbf577559c136ddd8232395" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:39.851ex; height:6.843ex;" alt="{\displaystyle \vartheta _{01}(x)=1-{\biggl \{}\sum _{k=1}^{\infty }2{\bigl [}x^{(2k-1)^{2}}-x^{(2k)^{2}}{\bigr ]}{\biggr \}}}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \vartheta _{01}(x)=1-2x+2x^{4}-2x^{9}+2x^{16}-2x^{25}\pm ...}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ϑ<!-- ϑ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>01</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1</mn> <mo>−<!-- − --></mo> <mn>2</mn> <mi>x</mi> <mo>+</mo> <mn>2</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>−<!-- − --></mo> <mn>2</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>9</mn> </mrow> </msup> <mo>+</mo> <mn>2</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>16</mn> </mrow> </msup> <mo>−<!-- − --></mo> <mn>2</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>25</mn> </mrow> </msup> <mo>±<!-- ± --></mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \vartheta _{01}(x)=1-2x+2x^{4}-2x^{9}+2x^{16}-2x^{25}\pm ...}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7d4a4fb8841d71cdf8d413b6a02c431752bc2947" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:47.696ex; height:3.176ex;" alt="{\displaystyle \vartheta _{01}(x)=1-2x+2x^{4}-2x^{9}+2x^{16}-2x^{25}\pm ...}"></span></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Identitäten_über_die_Oberpartitionsfunktion"><span id="Identit.C3.A4ten_.C3.BCber_die_Oberpartitionsfunktion"></span>Identitäten über die Oberpartitionsfunktion</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Partitionsfunktion&veaction=edit&section=34" title="Abschnitt bearbeiten: Identitäten über die Oberpartitionsfunktion" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Partitionsfunktion&action=edit&section=34" title="Quellcode des Abschnitts bearbeiten: Identitäten über die Oberpartitionsfunktion"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Über die Thetafunktion ϑ₀₁(x) gilt diese Identität: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \vartheta _{01}(x)^{-1}=(x;x)_{\infty }^{-1}(x;x^{2})_{\infty }^{-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ϑ<!-- ϑ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>01</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−<!-- − --></mo> <mn>1</mn> </mrow> </msup> <mo>=</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>;</mo> <mi>x</mi> <msubsup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞<!-- ∞ --></mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−<!-- − --></mo> <mn>1</mn> </mrow> </msubsup> <mo stretchy="false">(</mo> <mi>x</mi> <mo>;</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msubsup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞<!-- ∞ --></mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−<!-- − --></mo> <mn>1</mn> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \vartheta _{01}(x)^{-1}=(x;x)_{\infty }^{-1}(x;x^{2})_{\infty }^{-1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/91f9c2354e8527e4fe5662c5177cec5c68513251" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.545ex; height:3.176ex;" alt="{\displaystyle \vartheta _{01}(x)^{-1}=(x;x)_{\infty }^{-1}(x;x^{2})_{\infty }^{-1}}"></span></dd></dl> <p>Daraus folgt diese Formel: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\biggl [}\sum _{n=0}^{\infty }{\overline {P}}(n)x^{n}{\biggr ]}={\biggl [}\sum _{n=0}^{\infty }P(n)x^{n}{\biggr ]}{\biggl [}\sum _{n=0}^{\infty }Q(n)x^{n}{\biggr ]}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">[</mo> </mrow> </mrow> <munderover> <mo>∑<!-- ∑ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞<!-- ∞ --></mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>P</mi> <mo accent="false">¯<!-- ¯ --></mo> </mover> </mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">]</mo> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">[</mo> </mrow> </mrow> <munderover> <mo>∑<!-- ∑ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞<!-- ∞ --></mi> </mrow> </munderover> <mi>P</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">]</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">[</mo> </mrow> </mrow> <munderover> <mo>∑<!-- ∑ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞<!-- ∞ --></mi> </mrow> </munderover> <mi>Q</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\biggl [}\sum _{n=0}^{\infty }{\overline {P}}(n)x^{n}{\biggr ]}={\biggl [}\sum _{n=0}^{\infty }P(n)x^{n}{\biggr ]}{\biggl [}\sum _{n=0}^{\infty }Q(n)x^{n}{\biggr ]}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3321ec95e1b404a0019a3afce8addd07a40dce10" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:45.716ex; height:6.843ex;" alt="{\displaystyle {\biggl [}\sum _{n=0}^{\infty }{\overline {P}}(n)x^{n}{\biggr ]}={\biggl [}\sum _{n=0}^{\infty }P(n)x^{n}{\biggr ]}{\biggl [}\sum _{n=0}^{\infty }Q(n)x^{n}{\biggr ]}}"></span></dd></dl> <p>Und deswegen ist auch jene Formel für die Synthese der Zahlenfolge <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overline {P}}(n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>P</mi> <mo accent="false">¯<!-- ¯ --></mo> </mover> </mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {P}}(n)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8873a539ad8a7faddae443059099f07b192c9995" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.229ex; height:3.509ex;" alt="{\displaystyle {\overline {P}}(n)}"></span> gültig: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overline {P}}(n)=\sum _{k=0}^{n}P(n-k)Q(k)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>P</mi> <mo accent="false">¯<!-- ¯ --></mo> </mover> </mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munderover> <mo>∑<!-- ∑ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mi>P</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−<!-- − --></mo> <mi>k</mi> <mo stretchy="false">)</mo> <mi>Q</mi> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {P}}(n)=\sum _{k=0}^{n}P(n-k)Q(k)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/28ce2778ac163928a5625334f733bfbc24b736d9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:25.929ex; height:7.009ex;" alt="{\displaystyle {\overline {P}}(n)=\sum _{k=0}^{n}P(n-k)Q(k)}"></span></dd></dl> <p>Im nun Folgenden werden auch hierfür zwei Beispiele akkurat ausgeführt: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overline {P}}(6)=\sum _{k=0}^{6}P(6-k)Q(k)=}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>P</mi> <mo accent="false">¯<!-- ¯ --></mo> </mover> </mrow> <mo stretchy="false">(</mo> <mn>6</mn> <mo stretchy="false">)</mo> <mo>=</mo> <munderover> <mo>∑<!-- ∑ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </munderover> <mi>P</mi> <mo stretchy="false">(</mo> <mn>6</mn> <mo>−<!-- − --></mo> <mi>k</mi> <mo stretchy="false">)</mo> <mi>Q</mi> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {P}}(6)=\sum _{k=0}^{6}P(6-k)Q(k)=}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c881ab5e2dbb19dbbe38f2a5f1d763a302ef0d35" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:27.918ex; height:7.509ex;" alt="{\displaystyle {\overline {P}}(6)=\sum _{k=0}^{6}P(6-k)Q(k)=}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle =P(6)Q(0)+P(5)Q(1)+P(4)Q(2)+P(3)Q(3)+P(2)Q(4)+P(1)Q(5)+P(0)Q(6)=}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>=</mo> <mi>P</mi> <mo stretchy="false">(</mo> <mn>6</mn> <mo stretchy="false">)</mo> <mi>Q</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>+</mo> <mi>P</mi> <mo stretchy="false">(</mo> <mn>5</mn> <mo stretchy="false">)</mo> <mi>Q</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>+</mo> <mi>P</mi> <mo stretchy="false">(</mo> <mn>4</mn> <mo stretchy="false">)</mo> <mi>Q</mi> <mo stretchy="false">(</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo>+</mo> <mi>P</mi> <mo stretchy="false">(</mo> <mn>3</mn> <mo stretchy="false">)</mo> <mi>Q</mi> <mo stretchy="false">(</mo> <mn>3</mn> <mo stretchy="false">)</mo> <mo>+</mo> <mi>P</mi> <mo stretchy="false">(</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mi>Q</mi> <mo stretchy="false">(</mo> <mn>4</mn> <mo stretchy="false">)</mo> <mo>+</mo> <mi>P</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mi>Q</mi> <mo stretchy="false">(</mo> <mn>5</mn> <mo stretchy="false">)</mo> <mo>+</mo> <mi>P</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mi>Q</mi> <mo stretchy="false">(</mo> <mn>6</mn> <mo stretchy="false">)</mo> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle =P(6)Q(0)+P(5)Q(1)+P(4)Q(2)+P(3)Q(3)+P(2)Q(4)+P(1)Q(5)+P(0)Q(6)=}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/255b179eef61cbbbcaaf8ed048d1907fbbccc961" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:88.64ex; height:2.843ex;" alt="{\displaystyle =P(6)Q(0)+P(5)Q(1)+P(4)Q(2)+P(3)Q(3)+P(2)Q(4)+P(1)Q(5)+P(0)Q(6)=}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle =11\times 1+7\times 1+5\times 1+3\times 2+2\times 2+1\times 3+1\times 4=40}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>=</mo> <mn>11</mn> <mo>×<!-- × --></mo> <mn>1</mn> <mo>+</mo> <mn>7</mn> <mo>×<!-- × --></mo> <mn>1</mn> <mo>+</mo> <mn>5</mn> <mo>×<!-- × --></mo> <mn>1</mn> <mo>+</mo> <mn>3</mn> <mo>×<!-- × --></mo> <mn>2</mn> <mo>+</mo> <mn>2</mn> <mo>×<!-- × --></mo> <mn>2</mn> <mo>+</mo> <mn>1</mn> <mo>×<!-- × --></mo> <mn>3</mn> <mo>+</mo> <mn>1</mn> <mo>×<!-- × --></mo> <mn>4</mn> <mo>=</mo> <mn>40</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle =11\times 1+7\times 1+5\times 1+3\times 2+2\times 2+1\times 3+1\times 4=40}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/666e1102cdfb081bda9918cc3e6d22042c6ff49d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:62.239ex; height:2.343ex;" alt="{\displaystyle =11\times 1+7\times 1+5\times 1+3\times 2+2\times 2+1\times 3+1\times 4=40}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overline {P}}(7)=\sum _{k=0}^{7}P(7-k)Q(k)=}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>P</mi> <mo accent="false">¯<!-- ¯ --></mo> </mover> </mrow> <mo stretchy="false">(</mo> <mn>7</mn> <mo stretchy="false">)</mo> <mo>=</mo> <munderover> <mo>∑<!-- ∑ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>7</mn> </mrow> </munderover> <mi>P</mi> <mo stretchy="false">(</mo> <mn>7</mn> <mo>−<!-- − --></mo> <mi>k</mi> <mo stretchy="false">)</mo> <mi>Q</mi> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {P}}(7)=\sum _{k=0}^{7}P(7-k)Q(k)=}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/02e32dceb4d5733116e97b9d159da475d9db019a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:27.918ex; height:7.509ex;" alt="{\displaystyle {\overline {P}}(7)=\sum _{k=0}^{7}P(7-k)Q(k)=}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle =P(7)Q(0)+P(6)Q(1)+P(5)Q(2)+P(4)Q(3)+P(3)Q(4)+P(2)Q(5)+P(1)Q(6)+P(0)Q(7)=}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>=</mo> <mi>P</mi> <mo stretchy="false">(</mo> <mn>7</mn> <mo stretchy="false">)</mo> <mi>Q</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>+</mo> <mi>P</mi> <mo stretchy="false">(</mo> <mn>6</mn> <mo stretchy="false">)</mo> <mi>Q</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>+</mo> <mi>P</mi> <mo stretchy="false">(</mo> <mn>5</mn> <mo stretchy="false">)</mo> <mi>Q</mi> <mo stretchy="false">(</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo>+</mo> <mi>P</mi> <mo stretchy="false">(</mo> <mn>4</mn> <mo stretchy="false">)</mo> <mi>Q</mi> <mo stretchy="false">(</mo> <mn>3</mn> <mo stretchy="false">)</mo> <mo>+</mo> <mi>P</mi> <mo stretchy="false">(</mo> <mn>3</mn> <mo stretchy="false">)</mo> <mi>Q</mi> <mo stretchy="false">(</mo> <mn>4</mn> <mo stretchy="false">)</mo> <mo>+</mo> <mi>P</mi> <mo stretchy="false">(</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mi>Q</mi> <mo stretchy="false">(</mo> <mn>5</mn> <mo stretchy="false">)</mo> <mo>+</mo> <mi>P</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mi>Q</mi> <mo stretchy="false">(</mo> <mn>6</mn> <mo stretchy="false">)</mo> <mo>+</mo> <mi>P</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mi>Q</mi> <mo stretchy="false">(</mo> <mn>7</mn> <mo stretchy="false">)</mo> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle =P(7)Q(0)+P(6)Q(1)+P(5)Q(2)+P(4)Q(3)+P(3)Q(4)+P(2)Q(5)+P(1)Q(6)+P(0)Q(7)=}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/44b953162f1636dd339d9201d7d2cea19f197016" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:101.007ex; height:2.843ex;" alt="{\displaystyle =P(7)Q(0)+P(6)Q(1)+P(5)Q(2)+P(4)Q(3)+P(3)Q(4)+P(2)Q(5)+P(1)Q(6)+P(0)Q(7)=}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle =15\times 1+11\times 1+7\times 1+5\times 2+3\times 2+2\times 3+1\times 4+1\times 5=64}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>=</mo> <mn>15</mn> <mo>×<!-- × --></mo> <mn>1</mn> <mo>+</mo> <mn>11</mn> <mo>×<!-- × --></mo> <mn>1</mn> <mo>+</mo> <mn>7</mn> <mo>×<!-- × --></mo> <mn>1</mn> <mo>+</mo> <mn>5</mn> <mo>×<!-- × --></mo> <mn>2</mn> <mo>+</mo> <mn>3</mn> <mo>×<!-- × --></mo> <mn>2</mn> <mo>+</mo> <mn>2</mn> <mo>×<!-- × --></mo> <mn>3</mn> <mo>+</mo> <mn>1</mn> <mo>×<!-- × --></mo> <mn>4</mn> <mo>+</mo> <mn>1</mn> <mo>×<!-- × --></mo> <mn>5</mn> <mo>=</mo> <mn>64</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle =15\times 1+11\times 1+7\times 1+5\times 2+3\times 2+2\times 3+1\times 4+1\times 5=64}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/70e14ad81e2942fba639cd1f102d96264eedc5c6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:71.407ex; height:2.343ex;" alt="{\displaystyle =15\times 1+11\times 1+7\times 1+5\times 2+3\times 2+2\times 3+1\times 4+1\times 5=64}"></span></dd></dl> <p>Die hier gezeigte Tabelle stellt die genannten drei Zahlenfolgen zusammen: </p> <table class="wikitable"> <tbody><tr> <th>n </th> <th>P(n) </th> <th>Q(n)</th> <th><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overline {P}}(n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>P</mi> <mo accent="false">¯<!-- ¯ --></mo> </mover> </mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {P}}(n)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8873a539ad8a7faddae443059099f07b192c9995" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.229ex; height:3.509ex;" alt="{\displaystyle {\overline {P}}(n)}"></span> </th></tr> <tr> <td>0 </td> <td>1 </td> <td>1</td> <td>1 </td></tr> <tr> <td>1 </td> <td>1 </td> <td>1</td> <td>2 </td></tr> <tr> <td>2 </td> <td>2 </td> <td>1</td> <td>4 </td></tr> <tr> <td>3 </td> <td>3 </td> <td>2</td> <td>8 </td></tr> <tr> <td>4 </td> <td>5 </td> <td>2</td> <td>14 </td></tr> <tr> <td>5 </td> <td>7 </td> <td>3</td> <td>24 </td></tr> <tr> <td>6 </td> <td>11 </td> <td>4 </td> <td>40 </td></tr> <tr> <td>7 </td> <td>15 </td> <td>5 </td> <td>64 </td></tr></tbody></table> <div class="mw-heading mw-heading2"><h2 id="Literatur">Literatur</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Partitionsfunktion&veaction=edit&section=35" title="Abschnitt bearbeiten: Literatur" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Partitionsfunktion&action=edit&section=35" title="Quellcode des Abschnitts bearbeiten: Literatur"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Jacobus_Hendricus_van_Lint" class="mw-redirect" title="Jacobus Hendricus van Lint">Jacobus Hendricus van Lint</a>, R. M. Wilson: <cite style="font-style:italic">A Course in Combinatorics</cite>. 2. Auflage. Cambridge University Press, Cambridge 2001, <a href="/wiki/Spezial:ISBN-Suche/0521803403" class="internal mw-magiclink-isbn">ISBN 0-521-80340-3</a>.<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Partitionsfunktion&rft.au=Jacobus+Hendricus+van+Lint%2C+R.+M.+Wilson&rft.btitle=A+Course+in+Combinatorics&rft.date=2001&rft.edition=2&rft.genre=book&rft.isbn=0521803403&rft.place=Cambridge&rft.pub=Cambridge+University+Press" style="display:none"> </span></li> <li><a href="/wiki/Derrick_Henry_Lehmer" title="Derrick Henry Lehmer">Derrick Henry Lehmer</a>: <cite style="font-style:italic">Two nonexistence theorems on partitions</cite>. In: <cite style="font-style:italic">Bulletin of the American Mathematical Society</cite>. Volume 52, <span style="white-space:nowrap">Nr.<span style="display:inline-block;width:.2em"> </span>6</span>, 1946, <span style="white-space:nowrap">S.<span style="display:inline-block;width:.2em"> </span>538–544</span>, <a href="/wiki/Digital_Object_Identifier" title="Digital Object Identifier">doi</a>:<span class="uri-handle" style="white-space:nowrap"><a rel="nofollow" class="external text" href="https://doi.org/10.1090/S0002-9904-1946-08605-X">10.1090/S0002-9904-1946-08605-X</a></span> (<a rel="nofollow" class="external text" href="http://projecteuclid.org/euclid.bams/1183509416">projecteuclid.org</a> [abgerufen am 18. Februar 2012]).<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rfr_id=info:sid/de.wikipedia.org:Partitionsfunktion&rft.atitle=Two+nonexistence+theorems+on+partitions&rft.au=Derrick+Henry+Lehmer&rft.date=1946&rft.doi=10.1090%2FS0002-9904-1946-08605-X&rft.genre=journal&rft.issue=6&rft.jtitle=Bulletin+of+the+American+Mathematical+Society&rft.pages=538-544&rft.volume=Volume+52" style="display:none"> </span></li> <li><a href="/wiki/John_Edensor_Littlewood" title="John Edensor Littlewood">John Edensor Littlewood</a>: <cite class="lang" lang="en" dir="auto" style="font-style:italic">A Mathematician’s Miscellany</cite>. Eine Entdeckungsreise. Methuen, London 1953, <a href="/wiki/Spezial:ISBN-Suche/3540423869" class="internal mw-magiclink-isbn">ISBN 3-540-42386-9</a>, <span style="white-space:nowrap">S.<span style="display:inline-block;width:.2em"> </span>84–90</span> (englisch, <a rel="nofollow" class="external text" href="http://www.archive.org/details/mathematiciansmi033496mbp">Volltext</a> in verschiedenen Dateiformaten [abgerufen am 15. Februar 2012] Littlewood erzählt in diesem Buch unter anderem über Hardys Zusammenarbeit mit Ramanujan und wie sie das Problem <i>Approximation der Partitionsfunktion</i> 1918 gelöst haben).<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Partitionsfunktion&rft.au=John+Edensor+Littlewood&rft.btitle=A+Mathematician%E2%80%99s+Miscellany&rft.date=1953&rft.genre=book&rft.isbn=3540423869&rft.pages=84-90&rft.place=London&rft.pub=Methuen" style="display:none"> </span></li> <li><a href="/wiki/Ji%C5%99%C3%AD_Matou%C5%A1ek" title="Jiří Matoušek">Jiří Matoušek</a>, <a href="/wiki/Jaroslav_Ne%C5%A1et%C5%99il" title="Jaroslav Nešetřil">Jaroslav Nešetřil</a>: <cite style="font-style:italic">Diskrete Mathematik</cite>. Eine Entdeckungsreise. Springer, Berlin / Heidelberg / New York usw. 2002, <a href="/wiki/Spezial:ISBN-Suche/3540423869" class="internal mw-magiclink-isbn">ISBN 3-540-42386-9</a>, 10.7 Zahlpartitionen (<a rel="nofollow" class="external text" href="http://d-nb.info/963555103/04">Inhaltsverzeichnis</a> [abgerufen am 8. Februar 2012] englisch: <cite class="lang" lang="en" dir="auto" style="font-style:italic">Invitation to Discrete Mathematics</cite>. Übersetzt von Hans Mielke, Lehrbuch, das wenig Vorkenntnisse – gehobene Schulmathematik bis 2. Semester Mathematikstudium – voraussetzt).<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abookitem&rfr_id=info:sid/de.wikipedia.org:Partitionsfunktion&rft.atitle=10.7+Zahlpartitionen&rft.au=Ji%C5%99%C3%AD+Matou%C5%A1ek%2C+Jaroslav+Ne%C5%A1et%C5%99il&rft.btitle=Diskrete+Mathematik&rft.date=2002&rft.genre=bookitem&rft.isbn=3540423869&rft.place=Berlin+%2F+Heidelberg+%2F+New+York+usw.&rft.pub=Springer" style="display:none"> </span></li> <li>Sylvie Corteel, Jeremy Lovejoy: <cite style="font-style:italic">Overpartitions</cite>. Versailles / Talence 2004, <span style="white-space:nowrap">S.<span style="display:inline-block;width:.2em"> </span>1–13</span> (<a rel="nofollow" class="external autonumber" href="https://lovejoy.perso.math.cnrs.fr/overpartitions.pdf">[1]</a> [PDF; abgerufen am 10. März 2022]).<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Partitionsfunktion&rft.au=Sylvie+Corteel%2C+Jeremy+Lovejoy&rft.btitle=Overpartitions&rft.date=2004&rft.genre=book&rft.pages=1-13&rft.place=Versailles+%2F+Talence" style="display:none"> </span></li></ul> <p><b>Zu den Anwendungen in der Gruppentheorie:</b> </p> <ul><li>Thomas W. Hungerford: <cite style="font-style:italic">Algebra</cite>. In: <cite style="font-style:italic">Graduate texts in mathematics</cite>. 8. korrigierte Auflage. <span style="white-space:nowrap">Nr.<span style="display:inline-block;width:.2em"> </span>73</span>. Springer, New York / Berlin / Singapore / Tokyo / Heidelberg / Barcelona / Budapest / Hong Kong / London / Milan / Paris / Santa Clara 1996, <a href="/wiki/Spezial:ISBN-Suche/3540905189" class="internal mw-magiclink-isbn">ISBN 3-540-90518-9</a>, I. Groups, II. The Structure of Groups, <span style="white-space:nowrap">S.<span style="display:inline-block;width:.2em"> </span>35–82</span> (<a rel="nofollow" class="external text" href="http://d-nb.info/949253235/04">Inhaltsverzeichnis</a> <a rel="nofollow" class="external text" href="http://www.filediva.com/file_hungerford%20algebra.html">filediva.com</a> [abgerufen am 15. Februar 2012]).<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rfr_id=info:sid/de.wikipedia.org:Partitionsfunktion&rft.atitle=Algebra&rft.au=Thomas+W.+Hungerford&rft.date=1996&rft.edition=8.+korrigierte&rft.genre=journal&rft.isbn=3540905189&rft.issue=73&rft.jtitle=Graduate+texts+in+mathematics&rft.pages=35-82&rft.place=New+York+%2F+Berlin+%2F+Singapore+%2F+Tokyo+%2F+Heidelberg+%2F+Barcelona+%2F+Budapest+%2F+Hong+Kong+%2F+London+%2F+Milan+%2F+Paris+%2F+Santa+Clara&rft.pub=Springer" style="display:none"> </span></li></ul> <div class="mw-heading mw-heading2"><h2 id="Weblinks">Weblinks</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Partitionsfunktion&veaction=edit&section=36" title="Abschnitt bearbeiten: Weblinks" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Partitionsfunktion&action=edit&section=36" title="Quellcode des Abschnitts bearbeiten: Weblinks"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Eric_Weisstein" title="Eric Weisstein">Eric W. Weisstein</a>: <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/PartitionFunctionP.html"><i>Partition Function P</i>.</a> In: <i><a href="/wiki/MathWorld" title="MathWorld">MathWorld</a></i> (englisch). Partitionsfunktion <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(n)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5e303d2c14cd399b6f52b468c9fd44a542bed422" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.949ex; height:2.843ex;" alt="{\displaystyle P(n)}"></span>.</li> <li><a href="/wiki/Eric_Weisstein" title="Eric Weisstein">Eric W. Weisstein</a>: <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/PartitionFunctionQ.html"><i>Partition Function Q</i>.</a> In: <i><a href="/wiki/MathWorld" title="MathWorld">MathWorld</a></i> (englisch). Partitionsfunktion <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q(n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Q</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q(n)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/39f1477b71cf9622d2d860e02dbbcffaeac2f13a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.042ex; height:2.843ex;" alt="{\displaystyle Q(n)}"></span>.</li> <li>Das Computeralgebraprogramm <a href="/wiki/Maple_(Software)" title="Maple (Software)">Maple</a> enthält im Paket <i>combinat</i> die Funktion <a rel="nofollow" class="external text" href="http://www.maplesoft.com/support/help/Maple/view.aspx?path=combinat/partition">partition(n)</a>, die alle Zahlpartitionen der endlichen Mengen <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{1,2,\ldots n\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mo>…<!-- … --></mo> <mi>n</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{1,2,\ldots n\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/efe3956d3268d5f98d28caf0858271e074106666" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.223ex; height:2.843ex;" alt="{\displaystyle \{1,2,\ldots n\}}"></span> erzeugt und die Funktion <a rel="nofollow" class="external text" href="http://www.maplesoft.com/support/help/Maple/view.aspx?path=combinat%2fnumbpart">numbpart(n)</a>, die den Wert <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(n)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5e303d2c14cd399b6f52b468c9fd44a542bed422" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.949ex; height:2.843ex;" alt="{\displaystyle P(n)}"></span> der Partitionsfunktion berechnet.</li></ul> <div class="mw-heading mw-heading2"><h2 id="Einzelnachweise">Einzelnachweise</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Partitionsfunktion&veaction=edit&section=37" title="Abschnitt bearbeiten: Einzelnachweise" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Partitionsfunktion&action=edit&section=37" title="Quellcode des Abschnitts bearbeiten: Einzelnachweise"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><a href="#cite_ref-1">↑</a></span> <span class="reference-text"><a href="/wiki/Florian_Scheck" title="Florian Scheck">Florian Scheck</a>: <cite style="font-style:italic">Theoretische Physik 5: Statistische Theorie der Wärme</cite>. Springer, 2008, <a href="/wiki/Spezial:ISBN-Suche/9783540798231" class="internal mw-magiclink-isbn">ISBN 978-3-540-79823-1</a>, <span style="white-space:nowrap">S.<span style="display:inline-block;width:.2em"> </span>98</span> (<a rel="nofollow" class="external text" href="https://books.google.de/books?id=WclrgbUvr7sC&pg=PA98#v=onepage">eingeschränkte Vorschau</a> in der Google-Buchsuche).<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Partitionsfunktion&rft.au=Florian+Scheck&rft.btitle=Theoretische+Physik+5%3A+Statistische+Theorie+der+W%C3%A4rme&rft.date=2008&rft.genre=book&rft.isbn=9783540798231&rft.pages=98&rft.pub=Springer" style="display:none"> </span></span> </li> <li id="cite_note-MaNes-2"><span class="mw-cite-backlink">↑ <sup><a href="#cite_ref-MaNes_2-0">a</a></sup> <sup><a href="#cite_ref-MaNes_2-1">b</a></sup> <sup><a href="#cite_ref-MaNes_2-2">c</a></sup> <sup><a href="#cite_ref-MaNes_2-3">d</a></sup> <sup><a href="#cite_ref-MaNes_2-4">e</a></sup> <sup><a href="#cite_ref-MaNes_2-5">f</a></sup></span> <span class="reference-text">Matoušek, Nešetřil (2002)</span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><a href="#cite_ref-3">↑</a></span> <span class="reference-text"><a href="/wiki/Eric_Weisstein" title="Eric Weisstein">Eric W. Weisstein</a>: <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/PartitionFunctionQ.html"><i>Partition Function Q</i>.</a> In: <i><a href="/wiki/MathWorld" title="MathWorld">MathWorld</a></i> (englisch). </span> </li> <li id="cite_note-Macdonald-4"><span class="mw-cite-backlink"><a href="#cite_ref-Macdonald_4-0">↑</a></span> <span class="reference-text"><a href="/wiki/Ian_Macdonald" title="Ian Macdonald">Ian Macdonald</a>: <cite style="font-style:italic">Symmetric functions and Hall polynomials</cite>. Hrsg.: Oxford University Press. 2. Auflage. New York, <a href="/wiki/Spezial:ISBN-Suche/9780198739128" class="internal mw-magiclink-isbn">ISBN 978-0-19-873912-8</a>, <span style="white-space:nowrap">S.<span style="display:inline-block;width:.2em"> </span>1</span>.<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Partitionsfunktion&rft.au=Ian+Macdonald&rft.btitle=Symmetric+functions+and+Hall+polynomials&rft.edition=2&rft.genre=book&rft.isbn=9780198739128&rft.pages=1&rft.place=New+York" style="display:none"> </span></span> </li> <li id="cite_note-Steger-5"><span class="mw-cite-backlink"><a href="#cite_ref-Steger_5-0">↑</a></span> <span class="reference-text"> <a href="/wiki/Angelika_Steger" title="Angelika Steger">Angelika Steger</a>: <cite style="font-style:italic">Diskrete Strukturen 1: Kombinatorik, Graphentheorie, Algebra</cite>. Springer, 2001, <a href="/wiki/Spezial:ISBN-Suche/9783540675976" class="internal mw-magiclink-isbn">ISBN 978-3-540-67597-6</a>, <span style="white-space:nowrap">S.<span style="display:inline-block;width:.2em"> </span>36</span>.<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Partitionsfunktion&rft.au=Angelika+Steger&rft.btitle=Diskrete+Strukturen+1%3A+Kombinatorik%2C+Graphentheorie%2C+Algebra&rft.date=2001&rft.genre=book&rft.isbn=9783540675976&rft.pages=36&rft.pub=Springer" style="display:none"> </span></span> </li> <li id="cite_note-Zimmermann-6"><span class="mw-cite-backlink"><a href="#cite_ref-Zimmermann_6-0">↑</a></span> <span class="reference-text"> <a href="/wiki/Karl-Heinz_Zimmermann" title="Karl-Heinz Zimmermann">Karl-Heinz Zimmermann</a>: <cite style="font-style:italic">Diskrete Mathematik</cite>. <a href="/wiki/Books_on_Demand" title="Books on Demand">Books on Demand</a>, 2006, <a href="/wiki/Spezial:ISBN-Suche/9783833455292" class="internal mw-magiclink-isbn">ISBN 978-3-8334-5529-2</a>, <span style="white-space:nowrap">S.<span style="display:inline-block;width:.2em"> </span>115</span>.<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Partitionsfunktion&rft.au=Karl-Heinz+Zimmermann&rft.btitle=Diskrete+Mathematik&rft.date=2006&rft.genre=book&rft.isbn=9783833455292&rft.pages=115&rft.pub=Books+on+Demand" style="display:none"> </span></span> </li> <li id="cite_note-7"><span class="mw-cite-backlink"><a href="#cite_ref-7">↑</a></span> <span class="reference-text">Littlewood, 1953</span> </li> <li id="cite_note-8"><span class="mw-cite-backlink"><a href="#cite_ref-8">↑</a></span> <span class="reference-text">J. H. Bruinier, K. Ono: Algebraic formulas for the coefficients of half-integral weight harmonic weak Maass forms, Advances in Mathematics, Band 246, Seiten 198–219, 2013; <a rel="nofollow" class="external text" href="https://arxiv.org/abs/1104.1182">Preprint</a> im <a href="/wiki/ArXiv" title="ArXiv">ArXiv</a>, 2011.</span> </li> <li id="cite_note-9"><span class="mw-cite-backlink"><a href="#cite_ref-9">↑</a></span> <span class="reference-text"><span class="cite"><i>Eulers Erbe – Mathematiker feiern Entdeckung in der Zahlentheorie.</i> In: <i>sueddeutsche.de.</i> <a href="/wiki/S%C3%BCddeutsche_Zeitung" title="Süddeutsche Zeitung">Süddeutsche Zeitung</a>, 27. Januar 2011, ehemals im <style data-mw-deduplicate="TemplateStyles:r235239667">.mw-parser-output .dewiki-iconexternal>a{background-position:center right;background-repeat:no-repeat}body.skin-minerva .mw-parser-output .dewiki-iconexternal>a{background-image:url("https://upload.wikimedia.org/wikipedia/commons/a/a4/OOjs_UI_icon_external-link-ltr-progressive.svg")!important;background-size:10px;padding-right:13px!important}body.skin-timeless .mw-parser-output .dewiki-iconexternal>a,body.skin-monobook .mw-parser-output .dewiki-iconexternal>a{background-image:url("https://upload.wikimedia.org/wikipedia/commons/3/30/MediaWiki_external_link_icon.svg")!important;padding-right:13px!important}body.skin-vector .mw-parser-output .dewiki-iconexternal>a{background-image:url("https://upload.wikimedia.org/wikipedia/commons/9/96/Link-external-small-ltr-progressive.svg")!important;background-size:0.857em;padding-right:1em!important}</style><span class="dewiki-iconexternal"><a class="external text" href="https://redirecter.toolforge.org/?url=http%3A%2F%2Fwww.sueddeutsche.de%2FK5x383%2F3864679%2FEulers-Erbe.html">Original</a></span> (nicht mehr online verfügbar)<span>;</span><span class="Abrufdatum"> abgerufen am 8. März 2024</span>.<span style="display:none"><a rel="nofollow" class="external text" href="http://deadurl.invalid/http://www.sueddeutsche.de/K5x383/3864679/Eulers-Erbe.html">@1</a></span><span style="display:none"><a rel="nofollow" class="external text" href="http://www.sueddeutsche.de/K5x383/3864679/Eulers-Erbe.html">@2</a></span><span style="display:none"><a href="/w/index.php?title=Vorlage:Toter_Link/www.sueddeutsche.de&action=edit&redlink=1" class="new" title="Vorlage:Toter Link/www.sueddeutsche.de (Seite nicht vorhanden)">Vorlage:Toter Link/www.sueddeutsche.de</a></span> <small>(<a href="/wiki/Wikipedia:Defekte_Weblinks" title="Wikipedia:Defekte Weblinks">Seite nicht mehr abrufbar</a>. <a rel="nofollow" class="external text" href="http://timetravel.mementoweb.org/list/2010/http://www.sueddeutsche.de/K5x383/3864679/Eulers-Erbe.html">Suche in Webarchiven</a>)</small></span><span style="display: none;" class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Adc&rfr_id=info%3Asid%2Fde.wikipedia.org%3APartitionsfunktion&rft.title=Eulers+Erbe+%E2%80%93+Mathematiker+feiern+Entdeckung+in+der+Zahlentheorie&rft.description=Eulers+Erbe+%E2%80%93+Mathematiker+feiern+Entdeckung+in+der+Zahlentheorie&rft.identifier=http%3A%2F%2Fwww.sueddeutsche.de%2FK5x383%2F3864679%2FEulers-Erbe.html&rft.publisher=%5B%5BS%C3%BCddeutsche+Zeitung%5D%5D&rft.date=2011-01-27"> </span></span> </li> <li id="cite_note-10"><span class="mw-cite-backlink"><a href="#cite_ref-10">↑</a></span> <span class="reference-text">Ferrers war ein britischer Mathematiker (11. August 1829 bis 31. Januar 1903), siehe Matoušek, Nešetřil (2002)</span> </li> <li id="cite_note-11"><span class="mw-cite-backlink"><a href="#cite_ref-11">↑</a></span> <span class="reference-text">Ulrik Brandes: <a rel="nofollow" class="external text" href="http://www.inf.uni-konstanz.de/algo/lehre/ss08/mna/skript/einfuehrung.pdf"><i>Methoden der Netzwerkanalyse</i> – Vorlesungsskript, 1.15</a> (PDF; 316 kB) Universität Konstanz; abgerufen am 17. Februar 2012.</span> </li> <li id="cite_note-12"><span class="mw-cite-backlink"><a href="#cite_ref-12">↑</a></span> <span class="reference-text"><a href="/wiki/Leonhard_Euler" title="Leonhard Euler">Leonhard Euler</a>: <i>Introductio analysin infinitorum</i>, Band 1. Lausanne 1748, S. 253–275</span> </li> <li id="cite_note-Lehmer-13"><span class="mw-cite-backlink">↑ <sup><a href="#cite_ref-Lehmer_13-0">a</a></sup> <sup><a href="#cite_ref-Lehmer_13-1">b</a></sup></span> <span class="reference-text">Lehmer, 1946</span> </li> <li id="cite_note-14"><span class="mw-cite-backlink"><a href="#cite_ref-14">↑</a></span> <span class="reference-text"><span class="cite"><a rel="nofollow" class="external text" href="https://codegolf.stackexchange.com/questions/71941/strict-partitions-of-a-positive-integer"><i>code golf - Strict partitions of a positive integer.</i></a><span class="Abrufdatum"> Abgerufen am 9. März 2022</span>.</span><span style="display: none;" class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Adc&rfr_id=info%3Asid%2Fde.wikipedia.org%3APartitionsfunktion&rft.title=code+golf+-+Strict+partitions+of+a+positive+integer&rft.description=code+golf+-+Strict+partitions+of+a+positive+integer&rft.identifier=https%3A%2F%2Fcodegolf.stackexchange.com%2Fquestions%2F71941%2Fstrict-partitions-of-a-positive-integer"> </span></span> </li> <li id="cite_note-15"><span class="mw-cite-backlink"><a href="#cite_ref-15">↑</a></span> <span class="reference-text"><span class="cite"><a rel="nofollow" class="external text" href="https://oeis.org/A000009"><i>A000009 - OEIS.</i></a><span class="Abrufdatum"> Abgerufen am 9. März 2022</span>.</span><span style="display: none;" class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Adc&rfr_id=info%3Asid%2Fde.wikipedia.org%3APartitionsfunktion&rft.title=A000009+-+OEIS&rft.description=A000009+-+OEIS&rft.identifier=https%3A%2F%2Foeis.org%2FA000009"> </span></span> </li> <li id="cite_note-16"><span class="mw-cite-backlink"><a href="#cite_ref-16">↑</a></span> <span class="reference-text"><span class="cite">Eric W. Weisstein: <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/"><i>Partition Function Q.</i></a><span class="Abrufdatum"> Abgerufen am 9. März 2022</span> (englisch).</span><span style="display: none;" class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Adc&rfr_id=info%3Asid%2Fde.wikipedia.org%3APartitionsfunktion&rft.title=Partition+Function+Q&rft.description=Partition+Function+Q&rft.identifier=https%3A%2F%2Fmathworld.wolfram.com%2F&rft.creator=Eric+W.+Weisstein&rft.language=en"> </span></span> </li> </ol></div><!--esi <esi:include src="/esitest-fa8a495983347898/content" /> --><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1&useformat=desktop" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Abgerufen von „<a dir="ltr" href="https://de.wikipedia.org/w/index.php?title=Partitionsfunktion&oldid=244394086">https://de.wikipedia.org/w/index.php?title=Partitionsfunktion&oldid=244394086</a>“</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Wikipedia:Kategorien" title="Wikipedia:Kategorien">Kategorien</a>: <ul><li><a href="/wiki/Kategorie:Kombinatorik" title="Kategorie:Kombinatorik">Kombinatorik</a></li><li><a href="/wiki/Kategorie:Zahlentheorie" title="Kategorie:Zahlentheorie">Zahlentheorie</a></li></ul></div><div id="mw-hidden-catlinks" class="mw-hidden-catlinks mw-hidden-cats-hidden">Versteckte Kategorien: <ul><li><a href="/wiki/Kategorie:Wikipedia:Wikidata_P2812_verschieden" title="Kategorie:Wikipedia:Wikidata P2812 verschieden">Wikipedia:Wikidata P2812 verschieden</a></li><li><a href="/wiki/Kategorie:Wikipedia:Weblink_offline_deadurl" title="Kategorie:Wikipedia:Weblink offline deadurl">Wikipedia:Weblink offline deadurl</a></li></ul></div></div> </div> </div> <div id="mw-navigation"> <h2>Navigationsmenü</h2> <div id="mw-head"> <nav id="p-personal" class="mw-portlet mw-portlet-personal vector-user-menu-legacy vector-menu" aria-labelledby="p-personal-label" > <h3 id="p-personal-label" class="vector-menu-heading " > <span class="vector-menu-heading-label">Meine Werkzeuge</span> </h3> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-anonuserpage" class="mw-list-item"><span title="Benutzerseite der IP-Adresse, von der aus du Änderungen durchführst">Nicht angemeldet</span></li><li id="pt-anontalk" class="mw-list-item"><a href="/wiki/Spezial:Meine_Diskussionsseite" title="Diskussion über Änderungen von dieser IP-Adresse [n]" accesskey="n"><span>Diskussionsseite</span></a></li><li id="pt-anoncontribs" class="mw-list-item"><a href="/wiki/Spezial:Meine_Beitr%C3%A4ge" title="Eine Liste der Bearbeitungen, die von dieser IP-Adresse gemacht wurden [y]" accesskey="y"><span>Beiträge</span></a></li><li id="pt-createaccount" class="mw-list-item"><a href="/w/index.php?title=Spezial:Benutzerkonto_anlegen&returnto=Partitionsfunktion" title="Wir ermutigen dich dazu, ein Benutzerkonto zu erstellen und dich anzumelden. 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href="https://fa.wikipedia.org/wiki/%D8%AA%D8%A7%D8%A8%D8%B9_%D8%A7%D9%81%D8%B1%D8%A7%D8%B2_(%D9%86%D8%B8%D8%B1%DB%8C%D9%87_%D8%A7%D8%B9%D8%AF%D8%A7%D8%AF)" title="تابع افراز (نظریه اعداد) – Persisch" lang="fa" hreflang="fa" data-title="تابع افراز (نظریه اعداد)" data-language-autonym="فارسی" data-language-local-name="Persisch" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Funci%C3%B3n_de_partici%C3%B3n" title="Función de partición – Galicisch" lang="gl" hreflang="gl" data-title="Función de partición" data-language-autonym="Galego" data-language-local-name="Galicisch" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%A4%D7%95%D7%A0%D7%A7%D7%A6%D7%99%D7%99%D7%AA_%D7%94%D7%97%D7%9C%D7%95%D7%A7%D7%94_(%D7%AA%D7%95%D7%A8%D7%AA_%D7%94%D7%9E%D7%A1%D7%A4%D7%A8%D7%99%D7%9D)" title="פונקציית החלוקה (תורת המספרים) – Hebräisch" lang="he" hreflang="he" data-title="פונקציית החלוקה (תורת המספרים)" data-language-autonym="עברית" data-language-local-name="Hebräisch" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E5%88%86%E5%89%B2%E6%95%B0" title="分割数 – Japanisch" lang="ja" hreflang="ja" data-title="分割数" data-language-autonym="日本語" data-language-local-name="Japanisch" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-nl badge-Q70894304 mw-list-item" title=""><a href="https://nl.wikipedia.org/wiki/Partitiefunctie" title="Partitiefunctie – Niederländisch" lang="nl" hreflang="nl" data-title="Partitiefunctie" data-language-autonym="Nederlands" data-language-local-name="Niederländisch" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Fun%C3%A7%C3%A3o_de_parti%C3%A7%C3%A3o_(matem%C3%A1tica)" title="Função de partição (matemática) – Portugiesisch" lang="pt" hreflang="pt" data-title="Função de partição (matemática)" data-language-autonym="Português" data-language-local-name="Portugiesisch" class="interlanguage-link-target"><span>Português</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span class="wb-langlinks-edit wb-langlinks-link"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q15846551#sitelinks-wikipedia" title="Links auf Artikel in anderen Sprachen bearbeiten" class="wbc-editpage">Links bearbeiten</a></span></div> </div> </nav> </div> </div> <footer id="footer" class="mw-footer" > <ul id="footer-info"> <li id="footer-info-lastmod"> Diese Seite wurde zuletzt am 25. 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