Heltalspartition – Wikipedia

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class="vector-toc-numb">2</span> <span>Partitionsfunktionen</span> </div> </a> <button aria-controls="toc-Partitionsfunktionen-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Växla underavsnittet Partitionsfunktionen</span> </button> <ul id="toc-Partitionsfunktionen-sublist" class="vector-toc-list"> <li id="toc-Genererande_funktion" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Genererande_funktion"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>Genererande funktion</span> </div> </a> <ul id="toc-Genererande_funktion-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Kongruenser" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Kongruenser"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2</span> <span>Kongruenser</span> </div> </a> <ul id="toc-Kongruenser-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Approximationer" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Approximationer"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.3</span> <span>Approximationer</span> </div> </a> <ul id="toc-Approximationer-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-En_metod_att_beräkna_partitionsfunktionen" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#En_metod_att_beräkna_partitionsfunktionen"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.4</span> <span>En metod att beräkna partitionsfunktionen</span> </div> </a> <ul id="toc-En_metod_att_beräkna_partitionsfunktionen-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Källor" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Källor"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Källor</span> </div> </a> <ul id="toc-Källor-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Innehåll" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" title="Innehållsförteckning" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Växla innehållsförteckningen" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">Växla innehållsförteckningen</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Heltalspartition</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Gå till en artikel på ett annat språk. Tillgänglig på 21 språk" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-21" aria-hidden="true" ><span class="vector-icon mw-ui-icon-language-progressive mw-ui-icon-wikimedia-language-progressive"></span> <span class="vector-dropdown-label-text">21 språk</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%AA%D8%AC%D8%B2%D8%A6%D8%A9_(%D9%86%D8%B8%D8%B1%D9%8A%D8%A9_%D8%A7%D9%84%D8%A3%D8%B9%D8%AF%D8%A7%D8%AF)" title="تجزئة (نظرية الأعداد) – arabiska" lang="ar" hreflang="ar" data-title="تجزئة (نظرية الأعداد)" data-language-autonym="العربية" data-language-local-name="arabiska" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%A5%D0%B8%D1%81%D0%B5%D0%BF_%D0%BF%D0%B0%D0%B9%D0%BB%D0%B0%D0%B2%C4%95" title="Хисеп пайлавĕ – tjuvasjiska" lang="cv" hreflang="cv" data-title="Хисеп пайлавĕ" data-language-autonym="Чӑвашла" data-language-local-name="tjuvasjiska" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Integer_partition" title="Integer partition – engelska" lang="en" hreflang="en" data-title="Integer partition" data-language-autonym="English" data-language-local-name="engelska" class="interlanguage-link-target"><span>English</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Partici%C3%B3n_(teor%C3%ADa_de_n%C3%BAmeros)" title="Partición (teoría de números) – spanska" lang="es" hreflang="es" data-title="Partición (teoría de números)" data-language-autonym="Español" data-language-local-name="spanska" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%A7%D9%81%D8%B1%D8%A7%D8%B2_%D8%B9%D8%AF%D8%AF_%D8%B5%D8%AD%DB%8C%D8%AD" title="افراز عدد صحیح – persiska" lang="fa" hreflang="fa" data-title="افراز عدد صحیح" data-language-autonym="فارسی" data-language-local-name="persiska" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Partition_d%27un_entier" title="Partition d'un entier – franska" lang="fr" hreflang="fr" data-title="Partition d'un entier" data-language-autonym="Français" data-language-local-name="franska" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%9E%90%EC%97%B0%EC%88%98_%EB%B6%84%ED%95%A0" title="자연수 분할 – koreanska" lang="ko" hreflang="ko" data-title="자연수 분할" data-language-autonym="한국어" data-language-local-name="koreanska" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Partisi_(teori_bilangan)" title="Partisi (teori bilangan) – indonesiska" lang="id" hreflang="id" data-title="Partisi (teori bilangan)" data-language-autonym="Bahasa Indonesia" data-language-local-name="indonesiska" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Partizione_di_un_intero" title="Partizione di un intero – italienska" lang="it" hreflang="it" data-title="Partizione di un intero" data-language-autonym="Italiano" data-language-local-name="italienska" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%97%D7%9C%D7%95%D7%A7%D7%94_(%D7%A7%D7%95%D7%9E%D7%91%D7%99%D7%A0%D7%98%D7%95%D7%A8%D7%99%D7%A7%D7%94)" title="חלוקה (קומבינטוריקה) – hebreiska" lang="he" hreflang="he" data-title="חלוקה (קומבינטוריקה)" data-language-autonym="עברית" data-language-local-name="hebreiska" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Part%C3%ADci%C3%B3_(sz%C3%A1melm%C3%A9let)" title="Partíció (számelmélet) – ungerska" lang="hu" hreflang="hu" data-title="Partíció (számelmélet)" data-language-autonym="Magyar" data-language-local-name="ungerska" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Partitie_(getaltheorie)" title="Partitie (getaltheorie) – nederländska" lang="nl" hreflang="nl" data-title="Partitie (getaltheorie)" data-language-autonym="Nederlands" data-language-local-name="nederländska" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E8%87%AA%E7%84%B6%E6%95%B0%E3%81%AE%E5%88%86%E5%89%B2" title="自然数の分割 – japanska" lang="ja" hreflang="ja" data-title="自然数の分割" data-language-autonym="日本語" data-language-local-name="japanska" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%A0%D0%B0%D0%B7%D0%B1%D0%B8%D0%B5%D0%BD%D0%B8%D0%B5_%D1%87%D0%B8%D1%81%D0%BB%D0%B0" title="Разбиение числа – ryska" lang="ru" hreflang="ru" data-title="Разбиение числа" data-language-autonym="Русский" data-language-local-name="ryska" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Particioni_i_numrit_natyral" title="Particioni i numrit natyral – albanska" lang="sq" hreflang="sq" data-title="Particioni i numrit natyral" data-language-autonym="Shqip" data-language-local-name="albanska" class="interlanguage-link-target"><span>Shqip</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Particija_(teorija_%C5%A1tevil)" title="Particija (teorija števil) – slovenska" lang="sl" hreflang="sl" data-title="Particija (teorija števil)" data-language-autonym="Slovenščina" data-language-local-name="slovenska" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%8E%E0%AE%A3%E0%AF%8D_%E0%AE%AA%E0%AE%BF%E0%AE%B0%E0%AE%BF%E0%AE%B5%E0%AE%BF%E0%AE%A9%E0%AF%88" title="எண் பிரிவினை – tamil" lang="ta" hreflang="ta" data-title="எண் பிரிவினை" data-language-autonym="தமிழ்" data-language-local-name="tamil" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B8%81%E0%B8%B2%E0%B8%A3%E0%B9%81%E0%B8%9A%E0%B9%88%E0%B8%87%E0%B8%AA%E0%B9%88%E0%B8%A7%E0%B8%99_(%E0%B8%97%E0%B8%A4%E0%B8%A9%E0%B8%8E%E0%B8%B5%E0%B8%88%E0%B8%B3%E0%B8%99%E0%B8%A7%E0%B8%99)" title="การแบ่งส่วน (ทฤษฎีจำนวน) – thailändska" lang="th" hreflang="th" data-title="การแบ่งส่วน (ทฤษฎีจำนวน)" data-language-autonym="ไทย" data-language-local-name="thailändska" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%A0%D0%BE%D0%B7%D0%B1%D0%B8%D1%82%D1%82%D1%8F_%D1%87%D0%B8%D1%81%D0%BB%D0%B0" title="Розбиття числа – ukrainska" lang="uk" hreflang="uk" data-title="Розбиття числа" data-language-autonym="Українська" data-language-local-name="ukrainska" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/Ph%C3%A2n_ho%E1%BA%A1ch_(l%C3%BD_thuy%E1%BA%BFt_s%E1%BB%91)" title="Phân hoạch (lý thuyết số) – vietnamesiska" lang="vi" hreflang="vi" data-title="Phân hoạch (lý thuyết số)" data-language-autonym="Tiếng Việt" data-language-local-name="vietnamesiska" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E6%95%B4%E6%95%B8%E5%88%86%E6%8B%86" title="整數分拆 – kinesiska" lang="zh" hreflang="zh" data-title="整數分拆" data-language-autonym="中文" data-language-local-name="kinesiska" class="interlanguage-link-target"><span>中文</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/Q1082910#sitelinks-wikipedia" title="Redigera interwikilänkar" class="wbc-editpage">Redigera länkar</a></span></div> </div> </div> </div> </header> <div class="vector-page-toolbar"> <div class="vector-page-toolbar-container"> <div id="left-navigation"> <nav aria-label="Namnrymder"> <div id="p-associated-pages" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-associated-pages" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-nstab-main" class="selected vector-tab-noicon mw-list-item"><a href="/wiki/Heltalspartition" title="Visa innehållssidan [c]" accesskey="c"><span>Artikel</span></a></li><li id="ca-talk" class="vector-tab-noicon mw-list-item"><a href="/wiki/Diskussion:Heltalspartition" rel="discussion" title="Diskussion om innehållssidan [t]" accesskey="t"><span>Diskussion</span></a></li> </ul> </div> </div> <div id="vector-variants-dropdown" class="vector-dropdown emptyPortlet" > <input type="checkbox" id="vector-variants-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-variants-dropdown" class="vector-dropdown-checkbox " aria-label="Ändra språkvariant" > <label id="vector-variants-dropdown-label" for="vector-variants-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet" aria-hidden="true" ><span class="vector-dropdown-label-text">svenska</span> </label> <div class="vector-dropdown-content"> <div id="p-variants" class="vector-menu mw-portlet mw-portlet-variants emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> </div> </div> </nav> </div> <div id="right-navigation" class="vector-collapsible"> <nav aria-label="Visningar"> <div id="p-views" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-views" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-view" class="selected vector-tab-noicon mw-list-item"><a href="/wiki/Heltalspartition"><span>Läs</span></a></li><li id="ca-ve-edit" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=Heltalspartition&veaction=edit" title="Redigera denna sida [v]" accesskey="v"><span>Redigera</span></a></li><li id="ca-edit" class="collapsible vector-tab-noicon mw-list-item"><a href="/w/index.php?title=Heltalspartition&action=edit" title="Redigera wikitexten för den här sidan [e]" accesskey="e"><span>Redigera wikitext</span></a></li><li id="ca-history" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=Heltalspartition&action=history" title="Tidigare versioner av sidan [h]" accesskey="h"><span>Visa historik</span></a></li> </ul> </div> </div> </nav> <nav class="vector-page-tools-landmark" aria-label="Sidverktyg"> <div id="vector-page-tools-dropdown" class="vector-dropdown vector-page-tools-dropdown" > <input type="checkbox" id="vector-page-tools-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-tools-dropdown" class="vector-dropdown-checkbox " aria-label="Verktyg" > <label id="vector-page-tools-dropdown-label" for="vector-page-tools-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet" aria-hidden="true" ><span class="vector-dropdown-label-text">Verktyg</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-tools-unpinned-container" class="vector-unpinned-container"> <div id="vector-page-tools" class="vector-page-tools vector-pinnable-element"> <div class="vector-pinnable-header vector-page-tools-pinnable-header vector-pinnable-header-unpinned" data-feature-name="page-tools-pinned" data-pinnable-element-id="vector-page-tools" data-pinned-container-id="vector-page-tools-pinned-container" data-unpinned-container-id="vector-page-tools-unpinned-container" > <div class="vector-pinnable-header-label">Verktyg</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-page-tools.pin">flytta till sidofältet</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-page-tools.unpin">dölj</button> </div> <div id="p-cactions" class="vector-menu mw-portlet mw-portlet-cactions emptyPortlet vector-has-collapsible-items" title="Fler alternativ" > <div class="vector-menu-heading"> Åtgärder </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-more-view" class="selected vector-more-collapsible-item mw-list-item"><a href="/wiki/Heltalspartition"><span>Läs</span></a></li><li id="ca-more-ve-edit" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=Heltalspartition&veaction=edit" title="Redigera denna sida [v]" accesskey="v"><span>Redigera</span></a></li><li id="ca-more-edit" class="collapsible vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=Heltalspartition&action=edit" title="Redigera wikitexten för den här sidan [e]" accesskey="e"><span>Redigera wikitext</span></a></li><li id="ca-more-history" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=Heltalspartition&action=history"><span>Visa historik</span></a></li> </ul> </div> </div> <div id="p-tb" class="vector-menu mw-portlet mw-portlet-tb" > <div class="vector-menu-heading"> Allmänt </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="t-whatlinkshere" class="mw-list-item"><a href="/wiki/Special:L%C3%A4nkar_hit/Heltalspartition" title="Lista över alla wikisidor som länkar hit [j]" accesskey="j"><span>Sidor som länkar hit</span></a></li><li id="t-recentchangeslinked" class="mw-list-item"><a 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href="/w/index.php?title=Special:UrlShortener&url=https%3A%2F%2Fsv.wikipedia.org%2Fwiki%2FHeltalspartition"><span>Hämta förkortad url</span></a></li><li id="t-urlshortener-qrcode" class="mw-list-item"><a href="/w/index.php?title=Special:QrCode&url=https%3A%2F%2Fsv.wikipedia.org%2Fwiki%2FHeltalspartition"><span>Ladda ner QR-kod</span></a></li> </ul> </div> </div> <div id="p-coll-print_export" class="vector-menu mw-portlet mw-portlet-coll-print_export" > <div class="vector-menu-heading"> Skriv ut/exportera </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="coll-create_a_book" class="mw-list-item"><a href="/w/index.php?title=Special:Bok&bookcmd=book_creator&referer=Heltalspartition"><span>Skapa en bok</span></a></li><li id="coll-download-as-rl" class="mw-list-item"><a href="/w/index.php?title=Special:DownloadAsPdf&page=Heltalspartition&action=show-download-screen"><span>Ladda ned som PDF</span></a></li><li id="t-print" class="mw-list-item"><a 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class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-appearance.unpin">dölj</button> </div> </div> </div> </nav> </div> </div> <div id="bodyContent" class="vector-body" aria-labelledby="firstHeading" data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">Från Wikipedia</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="sv" dir="ltr"><dl class="noexcerpt dablink"><dd><i>För andra betydelser, se <a href="/wiki/Partition" class="mw-disambig" title="Partition">Partition</a>.</i></dd></dl> <p><b>Partition</b> av ett tal är, inom <a href="/wiki/Talteori" title="Talteori">talteori</a>, ett sätt att skriva ett positivt heltal n som en summa av positiva heltal utan hänsyn till termernas inbördes ordning. Ibland även kallad oordnad partition. <b>Partitionsfunktionen</b> <i>p</i>(<i>n</i>) ger antalet möjliga partitioner av talet <i>n</i>. Något enkelt sätt att beräkna <i>p</i>(<i>n</i>) finns inte. </p><p>Om hänsyn till termernas ordning tas, talar man om ordnade partitioner av <i>n</i>. Med uttrycket <i>k</i>-partition av talet <i>n</i> menas en partition av <i>n</i> som består av <i>k</i> termer. Det totala antalet ordnade partitioner av n är lika med <span class="mwe-math-element"><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^{n-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2^{n-1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/80c2cb3e3a7de902c9503fb34a17641df5896539" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.481ex; height:2.676ex;" alt="{\displaystyle 2^{n-1}}" /></span> och antalet ordnade k-partitioner av n är lika med <span class="mwe-math-element"><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 {n-1}{k-1}}}"> <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"> <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 {\binom {n-1}{k-1}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5e653c24bf6314397550312c50710ff03a66645c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:8.819ex; height:6.176ex;" alt="{\displaystyle {\binom {n-1}{k-1}}}" /></span>. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Exempel">Exempel</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Heltalspartition&veaction=edit&section=1" title="Redigera avsnitt: Exempel" class="mw-editsection-visualeditor"><span>redigera</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Heltalspartition&action=edit&section=1" title="Redigera avsnitts källkod: Exempel"><span>redigera wikitext</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Talet 5 kan partitioneras på 7 olika sätt: </p> <ul><li>5</li> <li>4 + 1</li> <li>3 + 2</li> <li>3 + 1 + 1</li> <li>2 + 2 + 1</li> <li>2 + 1 + 1 + 1</li> <li>1 + 1 + 1 + 1 + 1</li></ul> <p>Antalet 3-partitioner av talet 5 är lika med 2, antalet ordnade 3-partitioner av talet 5 är lika med 6 och det totala antalet ordnade partitioner av 5 är lika med 16. </p> <div class="mw-heading mw-heading2"><h2 id="Partitionsfunktionen">Partitionsfunktionen</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Heltalspartition&veaction=edit&section=2" title="Redigera avsnitt: Partitionsfunktionen" class="mw-editsection-visualeditor"><span>redigera</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Heltalspartition&action=edit&section=2" title="Redigera avsnitts källkod: Partitionsfunktionen"><span>redigera wikitext</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Partitionsfunktionen <i>p</i>(<i>n</i>) är en funktion, som ger antalet möjliga partitioner av <i>n</i>. Defininitionsmässigt är <i>p</i>(0) = 1. </p><p>De första värdena av partitionsfunktionen <i>p</i>(<i>n</i>) är: <i>p(</i>1<i>), </i>p<i>(</i>2<i>)... = </i> </p> <dl><dd>1, 2, 3, 5, 7, 11, 15, 22, 30, 42, ... (talföljd <a href="//oeis.org/A000041" class="extiw" title="oeis:A000041">A000041</a> i <a href="/wiki/N%C3%A4tuppslagsverket_%C3%B6ver_heltalsf%C3%B6ljder" title="Nätuppslagsverket över heltalsföljder">OEIS</a>)</dd></dl> <p>Vidare är <i>p</i>(100) = 190569292, <i>p</i>(1000) = 24061467864032622473692149727991 och </p> <dl><dd><i>p</i>(10000) = 36167251325636293988820471890953695495016030339315650422081868605887952568754066420592310556052906916435144.</dd></dl> <div class="mw-heading mw-heading3"><h3 id="Genererande_funktion">Genererande funktion</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Heltalspartition&veaction=edit&section=3" title="Redigera avsnitt: Genererande funktion" class="mw-editsection-visualeditor"><span>redigera</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Heltalspartition&action=edit&section=3" title="Redigera avsnitts källkod: Genererande funktion"><span>redigera wikitext</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Partitionsfunktionens <a href="/wiki/Genererande_funktion" title="Genererande funktion">genererande funktion</a> ges av </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 }\left({\frac {1}{1-x^{k}}}\right).}"> <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> <mo>(</mo> <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> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{n=0}^{\infty }p(n)x^{n}=\prod _{k=1}^{\infty }\left({\frac {1}{1-x^{k}}}\right).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/50f07d199eb8f2c86f45c8dcfabe3048acbb896c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:28.831ex; height:6.843ex;" alt="{\displaystyle \sum _{n=0}^{\infty }p(n)x^{n}=\prod _{k=1}^{\infty }\left({\frac {1}{1-x^{k}}}\right).}" /></span></dd></dl> <p>Genererande funktionen för <i>q</i>(<i>n</i>), antalet partitioner av <i>n</i> till olika delar, ges av </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 _{k=1}^{\infty }(1+x^{k})=\prod _{k=1}^{\infty }\left({\frac {1}{1-x^{2k-1}}}\right).}"> <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>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> <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> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <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> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{n=0}^{\infty }q(n)x^{n}=\prod _{k=1}^{\infty }(1+x^{k})=\prod _{k=1}^{\infty }\left({\frac {1}{1-x^{2k-1}}}\right).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eb40aae0265ddf737b9735b49dd4eda578bc919a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:45.952ex; height:6.843ex;" alt="{\displaystyle \sum _{n=0}^{\infty }q(n)x^{n}=\prod _{k=1}^{\infty }(1+x^{k})=\prod _{k=1}^{\infty }\left({\frac {1}{1-x^{2k-1}}}\right).}" /></span></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Kongruenser">Kongruenser</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Heltalspartition&veaction=edit&section=4" title="Redigera avsnitt: Kongruenser" class="mw-editsection-visualeditor"><span>redigera</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Heltalspartition&action=edit&section=4" title="Redigera avsnitts källkod: Kongruenser"><span>redigera wikitext</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/Srinivasa_Ramanujan" class="mw-redirect" title="Srinivasa Ramanujan">Srinivasa Ramanujan</a> upptäckte några <a href="/wiki/Kongruensrelation" title="Kongruensrelation">kongruenser</a> för partitionsfunktionen: </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(5k+4)\equiv 0{\pmod {5}}\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo stretchy="false">(</mo> <mn>5</mn> <mi>k</mi> <mo>+</mo> <mn>4</mn> <mo stretchy="false">)</mo> <mo>≡</mo> <mn>0</mn> <mrow class="MJX-TeXAtom-ORD"> <mspace width="1em"></mspace> <mo stretchy="false">(</mo> <mi>mod</mi> <mspace width="0.333em"></mspace> <mn>5</mn> <mo stretchy="false">)</mo> </mrow> <mspace width="thinmathspace"></mspace> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p(5k+4)\equiv 0{\pmod {5}}\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3f556a48ed47ed4728f66242d71cc31b271b1fdd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:24.939ex; height:2.843ex;" alt="{\displaystyle p(5k+4)\equiv 0{\pmod {5}}\,}" /></span></dd></dl> <p>Det här följer av en identitet av Ramanujan, </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 _{k=0}^{\infty }p(5k+4)x^{k}=5~{\frac {(x^{5})_{\infty }^{5}}{(x)_{\infty }^{6}}}}"> <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>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mi>p</mi> <mo stretchy="false">(</mo> <mn>5</mn> <mi>k</mi> <mo>+</mo> <mn>4</mn> <mo stretchy="false">)</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mo>=</mo> <mn>5</mn> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msup> <msubsup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msubsup> </mrow> <mrow> <mo stretchy="false">(</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>6</mn> </mrow> </msubsup> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{k=0}^{\infty }p(5k+4)x^{k}=5~{\frac {(x^{5})_{\infty }^{5}}{(x)_{\infty }^{6}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eb2f37eeb45d6d1e1c50ba798f2023c67a45e8b3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:27.262ex; height:7.176ex;" alt="{\displaystyle \sum _{k=0}^{\infty }p(5k+4)x^{k}=5~{\frac {(x^{5})_{\infty }^{5}}{(x)_{\infty }^{6}}}}" /></span></dd></dl> <p>där <span class="mwe-math-element"><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)_{\infty }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>x</mi> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x)_{\infty }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ac0b4aa05cc2b6537ce45f8c5d0095e1ab718c12" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.014ex; height:2.843ex;" alt="{\displaystyle (x)_{\infty }}" /></span> är <a href="/wiki/Q-Pochhammersymbolen" title="Q-Pochhammersymbolen">q-Pochhammersymbolen</a>, definierad som </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)_{\infty }=\prod _{m=1}^{\infty }(1-x^{m}).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>x</mi> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msub> <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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x)_{\infty }=\prod _{m=1}^{\infty }(1-x^{m}).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d925f35dc126b16e17a1d7364e4671a6672fbb98" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:21.12ex; height:6.843ex;" alt="{\displaystyle (x)_{\infty }=\prod _{m=1}^{\infty }(1-x^{m}).}" /></span></dd></dl> <p>Han upptäckte även kongruenser relaterade till 7 och 11: </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(7k+5)&\equiv 0{\pmod {7}}\\p(11k+6)&\equiv 0{\pmod {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>7</mn> <mi>k</mi> <mo>+</mo> <mn>5</mn> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>≡</mo> <mn>0</mn> <mrow class="MJX-TeXAtom-ORD"> <mspace width="1em"></mspace> <mo stretchy="false">(</mo> <mi>mod</mi> <mspace width="0.333em"></mspace> <mn>7</mn> <mo stretchy="false">)</mo> </mrow> </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> <mrow class="MJX-TeXAtom-ORD"> <mspace width="1em"></mspace> <mo stretchy="false">(</mo> <mi>mod</mi> <mspace width="0.333em"></mspace> <mn>11</mn> <mo stretchy="false">)</mo> </mrow> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}p(7k+5)&\equiv 0{\pmod {7}}\\p(11k+6)&\equiv 0{\pmod {11}}.\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59693cb765f5146b20ae5ce48adaeda19359940d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:28.186ex; height:6.176ex;" alt="{\displaystyle {\begin{aligned}p(7k+5)&\equiv 0{\pmod {7}}\\p(11k+6)&\equiv 0{\pmod {11}}.\end{aligned}}}" /></span></dd></dl> <p>och för <i>p=7</i> relationen </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 _{k=0}^{\infty }p(7k+5)x^{k}=7~{\frac {(x^{7})_{\infty }^{3}}{(x)_{\infty }^{4}}}+49~{\frac {(x^{7})_{\infty }^{7}}{(x)_{\infty }^{8}}}}"> <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>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mi>p</mi> <mo stretchy="false">(</mo> <mn>7</mn> <mi>k</mi> <mo>+</mo> <mn>5</mn> <mo stretchy="false">)</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mo>=</mo> <mn>7</mn> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>7</mn> </mrow> </msup> <msubsup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msubsup> </mrow> <mrow> <mo stretchy="false">(</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>4</mn> </mrow> </msubsup> </mrow> </mfrac> </mrow> <mo>+</mo> <mn>49</mn> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>7</mn> </mrow> </msup> <msubsup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>7</mn> </mrow> </msubsup> </mrow> <mrow> <mo stretchy="false">(</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>8</mn> </mrow> </msubsup> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{k=0}^{\infty }p(7k+5)x^{k}=7~{\frac {(x^{7})_{\infty }^{3}}{(x)_{\infty }^{4}}}+49~{\frac {(x^{7})_{\infty }^{7}}{(x)_{\infty }^{8}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/40f62e002774608153cd147dabf1c121d5a63a2c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:39.913ex; height:7.176ex;" alt="{\displaystyle \sum _{k=0}^{\infty }p(7k+5)x^{k}=7~{\frac {(x^{7})_{\infty }^{3}}{(x)_{\infty }^{4}}}+49~{\frac {(x^{7})_{\infty }^{7}}{(x)_{\infty }^{8}}}}" /></span></dd></dl> <p><a href="/w/index.php?title=A._O._L._Atkin&action=edit&redlink=1" class="new" title="A. O. L. Atkin [inte skriven än]">A. O. L. Atkin</a> har bevisat några andra kongruenser, såsom </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(11^{3}\cdot 13\cdot k+237)\equiv 0{\pmod {13}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo stretchy="false">(</mo> <msup> <mn>11</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>⋅</mo> <mn>13</mn> <mo>⋅</mo> <mi>k</mi> <mo>+</mo> <mn>237</mn> <mo stretchy="false">)</mo> <mo>≡</mo> <mn>0</mn> <mrow class="MJX-TeXAtom-ORD"> <mspace width="1em"></mspace> <mo stretchy="false">(</mo> <mi>mod</mi> <mspace width="0.333em"></mspace> <mn>13</mn> <mo stretchy="false">)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p(11^{3}\cdot 13\cdot k+237)\equiv 0{\pmod {13}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/19748f0e1d499b2829d71efef388ee9afeec2e5a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:36.586ex; height:3.176ex;" alt="{\displaystyle p(11^{3}\cdot 13\cdot k+237)\equiv 0{\pmod {13}}.}" /></span></dd></dl> <p>En något mer komplicerad kongruens av F. Johansson (2012) är </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(28995244292486005245947069k+28995221336976431135321047)\equiv 0{\pmod {29}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo stretchy="false">(</mo> <mn>28995244292486005245947069</mn> <mi>k</mi> <mo>+</mo> <mn>28995221336976431135321047</mn> <mo stretchy="false">)</mo> <mo>≡</mo> <mn>0</mn> <mrow class="MJX-TeXAtom-ORD"> <mspace width="1em"></mspace> <mo stretchy="false">(</mo> <mi>mod</mi> <mspace width="0.333em"></mspace> <mn>29</mn> <mo stretchy="false">)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p(28995244292486005245947069k+28995221336976431135321047)\equiv 0{\pmod {29}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d20796d1ba5d5bbe15b7ec3cfa9a33dea3ba26ba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:84.484ex; height:2.843ex;" alt="{\displaystyle p(28995244292486005245947069k+28995221336976431135321047)\equiv 0{\pmod {29}}.}" /></span></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Approximationer">Approximationer</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Heltalspartition&veaction=edit&section=5" title="Redigera avsnitt: Approximationer" class="mw-editsection-visualeditor"><span>redigera</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Heltalspartition&action=edit&section=5" title="Redigera avsnitts källkod: Approximationer"><span>redigera wikitext</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>En asymptotisk formel för <i>p</i>(<i>n</i>) är </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 {\frac {1}{4n{\sqrt {3}}}}\exp \left({\pi {\sqrt {\frac {2n}{3}}}}\right){\mbox{ då }}n\rightarrow \infty .}"> <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>4</mn> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>3</mn> </msqrt> </mrow> </mrow> </mfrac> </mrow> <mi>exp</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mrow> <mn>2</mn> <mi>n</mi> </mrow> <mn>3</mn> </mfrac> </msqrt> </mrow> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext> då </mtext> </mstyle> </mrow> <mi>n</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p(n)\sim {\frac {1}{4n{\sqrt {3}}}}\exp \left({\pi {\sqrt {\frac {2n}{3}}}}\right){\mbox{ då }}n\rightarrow \infty .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1ef3ba691f7c053e6690f3a0e11cb00a6754c4c8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; margin-left: -0.089ex; width:40.313ex; height:6.676ex;" alt="{\displaystyle p(n)\sim {\frac {1}{4n{\sqrt {3}}}}\exp \left({\pi {\sqrt {\frac {2n}{3}}}}\right){\mbox{ då }}n\rightarrow \infty .}" /></span></dd></dl> <p><a href="/wiki/G._H._Hardy" class="mw-redirect" title="G. H. Hardy">G. H. Hardy</a> och <a href="/wiki/Srinivasa_Aiyangar_Ramanujan" title="Srinivasa Aiyangar Ramanujan">Srinivasa Aiyangar Ramanujan</a> bevisade formeln 1918 och senare upptäcktes den oberoende av <a href="/w/index.php?title=J._V._Uspensky&action=edit&redlink=1" class="new" title="J. V. Uspensky [inte skriven än]">J. V. Uspensky</a> 1920. </p><p>Hardy and Ramanujan förbättrade senare formeln till </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}{2{\sqrt {2}}}}\sum _{k=1}^{v}{\sqrt {k}}\,A_{k}(n)\,{\frac {d}{dn}}\exp \left({\pi {\sqrt {\frac {2}{3}}}{\frac {\sqrt {n-{\frac {1}{24}}}}{k}}}\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>2</mn> <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>v</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>k</mi> </msqrt> </mrow> <mspace width="thinmathspace"></mspace> <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="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>d</mi> <mrow> <mi>d</mi> <mi>n</mi> </mrow> </mfrac> </mrow> <mi>exp</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mn>2</mn> <mn>3</mn> </mfrac> </msqrt> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msqrt> <mi>n</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>24</mn> </mfrac> </mrow> </msqrt> <mi>k</mi> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p(n)={\frac {1}{2{\sqrt {2}}}}\sum _{k=1}^{v}{\sqrt {k}}\,A_{k}(n)\,{\frac {d}{dn}}\exp \left({\pi {\sqrt {\frac {2}{3}}}{\frac {\sqrt {n-{\frac {1}{24}}}}{k}}}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/776b4ee8076ba303daa681c3734f15fdd68f9343" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.671ex; margin-left: -0.089ex; width:53.727ex; height:10.509ex;" alt="{\displaystyle p(n)={\frac {1}{2{\sqrt {2}}}}\sum _{k=1}^{v}{\sqrt {k}}\,A_{k}(n)\,{\frac {d}{dn}}\exp \left({\pi {\sqrt {\frac {2}{3}}}{\frac {\sqrt {n-{\frac {1}{24}}}}{k}}}\right)}" /></span></dd></dl> <p>där </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;\;(m,\,k)\,=\,1}e^{\pi i\left[s(m,\,k)\;-\;{\frac {1}{k}}2nm\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> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> <mspace width="thinmathspace"></mspace> <mo>≤</mo> <mspace width="thinmathspace"></mspace> <mi>m</mi> <mspace width="thinmathspace"></mspace> <mo><</mo> <mspace width="thinmathspace"></mspace> <mi>k</mi> <mo>;</mo> <mspace width="thickmathspace"></mspace> <mo stretchy="false">(</mo> <mi>m</mi> <mo>,</mo> <mspace width="thinmathspace"></mspace> <mi>k</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace"></mspace> <mo>=</mo> <mspace width="thinmathspace"></mspace> <mn>1</mn> </mrow> </munder> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>π</mi> <mi>i</mi> <mrow> <mo>[</mo> <mrow> <mi>s</mi> <mo stretchy="false">(</mo> <mi>m</mi> <mo>,</mo> <mspace width="thinmathspace"></mspace> <mi>k</mi> <mo stretchy="false">)</mo> <mspace width="thickmathspace"></mspace> <mo>−</mo> <mspace width="thickmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>k</mi> </mfrac> </mrow> <mn>2</mn> <mi>n</mi> <mi>m</mi> </mrow> <mo>]</mo> </mrow> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A_{k}(n)=\sum _{0\,\leq \,m\,<\,k;\;(m,\,k)\,=\,1}e^{\pi i\left[s(m,\,k)\;-\;{\frac {1}{k}}2nm\right]}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/46b2b3b8738670905f5695dd52afeef79ddb484d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:42.718ex; height:7.009ex;" alt="{\displaystyle A_{k}(n)=\sum _{0\,\leq \,m\,<\,k;\;(m,\,k)\,=\,1}e^{\pi i\left[s(m,\,k)\;-\;{\frac {1}{k}}2nm\right]}.}" /></span></dd></dl> <p>Här betyder (<i>m</i>, <i>n</i>) = 1 att summan går över alla värden på <i>m</i> <a href="/wiki/Relativt_prima" title="Relativt prima">relativt prima</a> till <i>n</i>. Funktionen <i>s</i>(<i>m</i>, <i>k</i>) är en <a href="/wiki/Dedekindsumma" title="Dedekindsumma">Dedekindsumma</a>. </p><p>1937 förbättrade <a href="/w/index.php?title=Hans_Rademacher&action=edit&redlink=1" class="new" title="Hans Rademacher [inte skriven än]">Hans Rademacher</a> Hardys och Ramanujans resultat genom att bevisa en konvergerande serie för <i>p</i>(<i>n</i>). Serien är </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 }{\sqrt {k}}\,A_{k}(n)\,{\frac {d}{dn}}\left({{\frac {1}{\sqrt {n-{\frac {1}{24}}}}}\sinh \left[{{\frac {\pi }{k}}{\sqrt {{\frac {2}{3}}\left(n-{\frac {1}{24}}\right)}}}\right]}\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</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> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>k</mi> </msqrt> </mrow> <mspace width="thinmathspace"></mspace> <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="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>d</mi> <mrow> <mi>d</mi> <mi>n</mi> </mrow> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msqrt> <mi>n</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>24</mn> </mfrac> </mrow> </msqrt> </mfrac> </mrow> <mi>sinh</mi> <mo>⁡</mo> <mrow> <mo>[</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>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> <mo>(</mo> <mrow> <mi>n</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>24</mn> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> </msqrt> </mrow> </mrow> <mo>]</mo> </mrow> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p(n)={\frac {1}{\pi {\sqrt {2}}}}\sum _{k=1}^{\infty }{\sqrt {k}}\,A_{k}(n)\,{\frac {d}{dn}}\left({{\frac {1}{\sqrt {n-{\frac {1}{24}}}}}\sinh \left[{{\frac {\pi }{k}}{\sqrt {{\frac {2}{3}}\left(n-{\frac {1}{24}}\right)}}}\right]}\right).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ea7e556558190367b9ee0b3fd50be842fa047fb4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.671ex; margin-left: -0.089ex; width:70.664ex; height:10.509ex;" alt="{\displaystyle p(n)={\frac {1}{\pi {\sqrt {2}}}}\sum _{k=1}^{\infty }{\sqrt {k}}\,A_{k}(n)\,{\frac {d}{dn}}\left({{\frac {1}{\sqrt {n-{\frac {1}{24}}}}}\sinh \left[{{\frac {\pi }{k}}{\sqrt {{\frac {2}{3}}\left(n-{\frac {1}{24}}\right)}}}\right]}\right).}" /></span></dd></dl> <div class="mw-heading mw-heading3"><h3 id="En_metod_att_beräkna_partitionsfunktionen"><span id="En_metod_att_ber.C3.A4kna_partitionsfunktionen"></span>En metod att beräkna partitionsfunktionen</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Heltalspartition&veaction=edit&section=6" title="Redigera avsnitt: En metod att beräkna partitionsfunktionen" class="mw-editsection-visualeditor"><span>redigera</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Heltalspartition&action=edit&section=6" title="Redigera avsnitts källkod: En metod att beräkna partitionsfunktionen"><span>redigera wikitext</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Låt <b>P(n,k)</b> beteckna antalet partitioner av heltalet <b>n</b> som består av <b>k</b> termer och skriv dem i en tabell med en rad för varje <b>n</b> och varje <b>k</b> i en kolonn, enligt nedan (översta raden motsvarar <b>n=0</b>): </p> <table align="center"> <tbody><tr> <td>1 </td></tr> <tr> <td>1 </td></tr> <tr> <td>1</td> <td>1 </td></tr> <tr> <td>1</td> <td>1</td> <td>1 </td></tr> <tr> <td>1</td> <td>2</td> <td>1</td> <td>1 </td></tr> <tr> <td>1</td> <td>2</td> <td>2</td> <td>1</td> <td>1 </td></tr> <tr> <td>1</td> <td>3</td> <td>3</td> <td>2</td> <td>1</td> <td>1 </td></tr> <tr> <td>1</td> <td>3</td> <td>4</td> <td>3</td> <td>2</td> <td>1</td> <td>1 </td></tr> <tr> <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> <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></tbody></table> <p>Självklart är summan av talen i varje rad lika med antalet partitioner av <b>n</b>. </p><p>Den här tabellen skapas genom att man för varje <b>k</b> i rad <b>n</b> bildar <b>P(n,k)</b> genom att lägga ihop de <b>k</b> talen längst till vänster i rad <b>n-k</b> (om <b>k>n-k</b>, "fyller man på" raden med nollor så långt det behövs åt höger). </p> <dl><dd><b>Exempel</b></dd> <dd>I den nedersta raden (<b>n=9</b>) är den första ettan lika med ettan längst till vänster i raden ovanför (<b>P(9,1)=P(8,1)=1</b>). Nästa tal, 4, är summan av de två första talen två rader ovanför (<b>P(9,2)=P(7,1)+P(7,2)=1+3=4</b>). Tredje talet, 7, är summan av de tre första talen tre rader ovanför (<b>P(9,3)=P(6,1)+P(6,2)+P(6,3)=1+3+3=7</b>). <b>P(9,4)=P(5,1)+P(5,2)+P(5,3)+P(5,4)=1+2+2+1=6</b>. <b>P(9,5)=P(4,1)+P(4,2)+P(4,3)+P(4,4)+"P(4,5)"=1+2+1+1+0=5</b>. Etcetera...</dd></dl> <p>Att man verkligen får värdena på <b>P(n,k)</b> på detta sätt inses om man beaktar att man måste ha exakt <b>k</b> termer som alla är större än noll. När vi tilldelat dessa <b>k</b> termer det minimala värdet ett återstår <b>n-k</b> att fördela på dessa <b>k</b> termer. Detta kan göras på det antal sätt som är lika med summan av <b>P(n-k,1)</b> till <b>P(n-k,k)</b> (vi kan fördela resten, <b>n-k</b>, på en av termerna på ett sätt, på två av termerna på <b>P(n-k,2)</b> sätt, etcetera, och när vi antingen når <b>n-k</b> finns inte mer "rest" kvar att fördela eller när vi når <b>k</b> finns det inte fler termer att fördela "resten" på). </p> <div class="mw-heading mw-heading2"><h2 id="Källor"><span id="K.C3.A4llor"></span>Källor</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Heltalspartition&veaction=edit&section=7" title="Redigera avsnitt: Källor" class="mw-editsection-visualeditor"><span>redigera</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Heltalspartition&action=edit&section=7" title="Redigera avsnitts källkod: Källor"><span>redigera wikitext</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/w/index.php?title=George_E._Andrews&action=edit&redlink=1" class="new" title="George E. Andrews [inte skriven än]">George E. Andrews</a>, <i>The Theory of Partitions</i> (1976), Cambridge University Press. <i> <a href="/wiki/Special:Bokk%C3%A4llor/0-521-63766-X" title="Special:Bokkällor/0-521-63766-X">ISBN 0-521-63766-X</a> </i>.</li> <li><cite style="font-style:normal" class="book" id="CITEREFApostol1990"><a href="/w/index.php?title=Tom_M._Apostol&action=edit&redlink=1" class="new" title="Tom M. Apostol [inte skriven än]">Apostol, Tom M.</a> (1990) [1976]. <i><span>Modular functions and Dirichlet series in number theory</span></i>. <a href="/w/index.php?title=Graduate_Texts_in_Mathematics&action=edit&redlink=1" class="new" title="Graduate Texts in Mathematics [inte skriven än]">Graduate Texts in Mathematics</a>. "41" (2nd). New York etc.: <a href="/wiki/Springer-Verlag" class="mw-redirect" title="Springer-Verlag">Springer-Verlag</a>. <a href="/wiki/Special:Bokk%C3%A4llor/0-387-97127-0" title="Special:Bokkällor/0-387-97127-0">ISBN 0-387-97127-0</a></cite><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Modular+functions+and+Dirichlet+series+in+number+theory&rft.aulast=Apostol&rft.aufirst=Tom+M.&rft.au=Apostol%2C+Tom+M.&rft.date=1990&rft.series=%5B%5BGraduate+Texts+in+Mathematics%5D%5D&rft.volume=41&rft.edition=2nd&rft.place=New+York+etc.&rft.pub=%5B%5BSpringer-Verlag%5D%5D&rft.isbn=0-387-97127-0&rfr_id=info:sid/en.wikipedia.org:Heltalspartition"><span style="display: none;"> </span></span> <i>(See chapter 5 for a modern pedagogical intro to Rademacher's formula)</i>.</li> <li><a href="/w/index.php?title=Mall:Hardy_and_Wright&action=edit&redlink=1" class="new" title="Mall:Hardy and Wright [inte skriven än]">Mall:Hardy and Wright</a></li> <li><cite style="font-style:normal" class="journal" id="CITEREFLehmer1939"><a href="/w/index.php?title=D._H._Lehmer&action=edit&redlink=1" class="new" title="D. H. Lehmer [inte skriven än]">Lehmer, D. H.</a> (1939). ”On the remainder and convergence of the series for the partition function”. <i>Trans. Amer. Math. Soc.</i> 46: sid. 362–373. <a href="/wiki/Digital_object_identifier" title="Digital object identifier">doi</a>:<span class="neverexpand"><a rel="nofollow" class="external text" href="https://dx.doi.org/10.1090%2FS0002-9947-1939-0000410-9">10.1090/S0002-9947-1939-0000410-9</a></span>.</cite><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=On+the+remainder+and+convergence+of+the+series+for+the+partition+function&rft.jtitle=Trans.+Amer.+Math.+Soc.&rft.aulast=Lehmer&rft.aufirst=D.+H.&rft.au=Lehmer%2C+D.+H.&rft.date=1939&rft.volume=46&rft.pages=sid.%26nbsp%3B362%E2%80%93373&rft_id=info:doi/10.1090%2FS0002-9947-1939-0000410-9&rfr_id=info:sid/en.wikipedia.org:Heltalspartition"><span style="display: none;"> </span></span> Provides the main formula (no derivatives), remainder, and older form for A<sub>k</sub>(n).)<i></i></li> <li>Gupta, Gwyther, Miller, <i>Roy. Soc. Math. Tables, vol 4, Tables of partitions</i>, (1962) <i>(Has text, nearly complete bibliography, but they (and Abramowitz) missed the Selberg formula for A<sub>k</sub>(n), which is in Whiteman.)</i></li> <li><cite style="font-style:normal" class="book" id="CITEREFMacdonald1979"><a href="/w/index.php?title=Ian_G._Macdonald&action=edit&redlink=1" class="new" title="Ian G. Macdonald [inte skriven än]">Macdonald, Ian G.</a> (1979). <i><span>Symmetric functions and Hall polynomials</span></i>. Oxford Mathematical Monographs. <a href="/wiki/Oxford_University_Press" title="Oxford University Press">Oxford University Press</a>. <a href="/wiki/Special:Bokk%C3%A4llor/0-19-853530-9" title="Special:Bokkällor/0-19-853530-9">ISBN 0-19-853530-9</a></cite><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Symmetric+functions+and+Hall+polynomials&rft.aulast=Macdonald&rft.aufirst=Ian+G.&rft.au=Macdonald%2C+Ian+G.&rft.date=1979&rft.series=Oxford+Mathematical+Monographs&rft.pub=%5B%5BOxford+University+Press%5D%5D&rft.isbn=0-19-853530-9&rfr_id=info:sid/en.wikipedia.org:Heltalspartition"><span style="display: none;"> </span></span> (See section I.1)</li> <li><cite style="font-style:normal" class="book" id="CITEREFNathanson2000">Nathanson, M.B. (2000). <i><span>Elementary Methods in Number Theory</span></i>. Graduate Texts in Mathematics. "195". <a href="/wiki/Springer-Verlag" class="mw-redirect" title="Springer-Verlag">Springer-Verlag</a>. <a href="/wiki/Special:Bokk%C3%A4llor/0-387-98912-9" title="Special:Bokkällor/0-387-98912-9">ISBN 0-387-98912-9</a></cite><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Elementary+Methods+in+Number+Theory&rft.aulast=Nathanson&rft.aufirst=M.B.&rft.au=Nathanson%2C+M.B.&rft.date=2000&rft.series=Graduate+Texts+in+Mathematics&rft.volume=195&rft.pub=%5B%5BSpringer-Verlag%5D%5D&rft.isbn=0-387-98912-9&rfr_id=info:sid/en.wikipedia.org:Heltalspartition"><span style="display: none;"> </span></span></li> <li><a href="/w/index.php?title=Ken_Ono&action=edit&redlink=1" class="new" title="Ken Ono [inte skriven än]">Ken Ono</a>, <i>Distribution of the partition function modulo m</i>, Annals of Mathematics <b>151</b> (2000) pp 293–307.<i> (This paper proves congruences modulo every prime greater than 3)</i></li> <li><a href="/w/index.php?title=Marcus_du_Sautoy&action=edit&redlink=1" class="new" title="Marcus du Sautoy [inte skriven än]">Sautoy, Marcus Du.</a> The Music of the Primes. New York: Perennial-HarperCollins, 2003.</li> <li><a href="/w/index.php?title=Richard_P._Stanley&action=edit&redlink=1" class="new" title="Richard P. Stanley [inte skriven än]">Richard P. Stanley</a>, <a rel="nofollow" class="external text" href="http://www-math.mit.edu/~rstan/ec/"><i>Enumerative Combinatorics</i>, Volumes 1 and 2</a>. Cambridge University Press, 1999 <a href="/wiki/Special:Bokk%C3%A4llor/0-521-56069-1" title="Special:Bokkällor/0-521-56069-1">ISBN 0-521-56069-1</a></li> <li><cite style="font-style:normal" class="book" id="CITEREFWhiteman1956">Whiteman, A. L. (1956). <i><a rel="nofollow" class="external text" href="https://web.archive.org/web/20070312053718/http://projecteuclid.org/Dienst/UI/1.0/Summarize/euclid.pjm/1103044252">A sum connected with the series for the partition function</a></i>. "6". sid. 159–176. Arkiverad från <a rel="nofollow" class="external text" href="http://projecteuclid.org/Dienst/UI/1.0/Summarize/euclid.pjm/1103044252">originalet</a> den 12 mars 2007<span class="printonly">. <a rel="nofollow" class="external free" href="https://web.archive.org/web/20070312053718/http://projecteuclid.org/Dienst/UI/1.0/Summarize/euclid.pjm/1103044252">https://web.archive.org/web/20070312053718/http://projecteuclid.org/Dienst/UI/1.0/Summarize/euclid.pjm/1103044252</a></span><span class="reference-accessdate">. Läst 9 december 2013</span></cite><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+sum+connected+with+the+series+for+the+partition+function&rft.aulast=Whiteman&rft.aufirst=A.+L.&rft.au=Whiteman%2C+A.+L.&rft.date=1956&rft.volume=6&rft.issue=1&rft.pages=sid.%26nbsp%3B159%E2%80%93176&rft_id=https%3A%2F%2Fweb.archive.org%2Fweb%2F20070312053718%2Fhttp%3A%2F%2Fprojecteuclid.org%2FDienst%2FUI%2F1.0%2FSummarize%2Feuclid.pjm%2F1103044252&rfr_id=info:sid/en.wikipedia.org:Heltalspartition"><span style="display: none;"> </span></span> <i>(Provides the Selberg formula. The older form is the finite Fourier expansion of Selberg.)</i></li> <li><a href="/w/index.php?title=Hans_Rademacher&action=edit&redlink=1" class="new" title="Hans Rademacher [inte skriven än]">Hans Rademacher</a>, <i>Collected Papers of Hans Rademacher</i>, (1974) MIT Press; v II, p 100–107, 108–122, 460–475.</li> <li><cite style="font-style:normal" class="book" id="CITEREFMiklós_Bóna2002"><a href="/w/index.php?title=Mikl%C3%B3s_B%C3%B3na&action=edit&redlink=1" class="new" title="Miklós Bóna [inte skriven än]">Miklós Bóna</a> (2002). <i><span>A Walk Through Combinatorics: An Introduction to Enumeration and Graph Theory</span></i>. World Scientific Publishing. <a href="/wiki/Special:Bokk%C3%A4llor/981-02-4900-4" title="Special:Bokkällor/981-02-4900-4">ISBN 981-02-4900-4</a></cite><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+Walk+Through+Combinatorics%3A+An+Introduction+to+Enumeration+and+Graph+Theory&rft.aulast=Mikl%C3%B3s+B%C3%B3na&rft.au=Mikl%C3%B3s+B%C3%B3na&rft.date=2002&rft.pub=World+Scientific+Publishing&rft.isbn=981-02-4900-4&rfr_id=info:sid/en.wikipedia.org:Heltalspartition"><span style="display: none;"> </span></span> (qn elementary introduction to the topic of integer partition, including a discussion of Ferrers graphs)</li> <li><cite style="font-style:normal" class="book" id="CITEREFGeorge_E._Andrews,_Kimmo_Eriksson2004">George E. Andrews, Kimmo Eriksson (2004). <i><span>Integer Partitions</span></i>. Cambridge University Press. <a href="/wiki/Special:Bokk%C3%A4llor/0-521-60090-1" title="Special:Bokkällor/0-521-60090-1">ISBN 0-521-60090-1</a></cite><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Integer+Partitions&rft.aulast=George+E.+Andrews%2C+Kimmo+Eriksson&rft.au=George+E.+Andrews%2C+Kimmo+Eriksson&rft.date=2004&rft.pub=Cambridge+University+Press&rft.isbn=0-521-60090-1&rfr_id=info:sid/en.wikipedia.org:Heltalspartition"><span style="display: none;"> </span></span></li> <li>'A Disappearing Number', devised piece by <a href="/w/index.php?title=Complicite&action=edit&redlink=1" class="new" title="Complicite [inte skriven än]">Complicite</a>, mention Ramanujan's work on the Partition Function, 2007</li></ul>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?useformat=desktop&type=1x1&usesul3=0" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Hämtad från ”<a dir="ltr" href="https://sv.wikipedia.org/w/index.php?title=Heltalspartition&oldid=49005005">https://sv.wikipedia.org/w/index.php?title=Heltalspartition&oldid=49005005</a>”</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Wikipedia:Kategorier" title="Wikipedia:Kategorier">Kategorier</a>: <ul><li><a href="/wiki/Kategori:Kombinatorik" title="Kategori:Kombinatorik">Kombinatorik</a></li><li><a href="/wiki/Kategori:Aritmetiska_funktioner" title="Kategori:Aritmetiska funktioner">Aritmetiska funktioner</a></li><li><a href="/wiki/Kategori:Heltalsm%C3%A4ngder" title="Kategori:Heltalsmängder">Heltalsmängder</a></li></ul></div><div id="mw-hidden-catlinks" class="mw-hidden-catlinks mw-hidden-cats-hidden">Dold kategori: <ul><li><a href="/wiki/Kategori:Wikipedia:Projekt_%C3%B6vers%C3%A4tta_k%C3%A4llmallar" title="Kategori:Wikipedia:Projekt översätta källmallar">Wikipedia:Projekt översätta källmallar</a></li></ul></div></div> </div> </main> </div> <div class="mw-footer-container"> <footer id="footer" class="mw-footer" > <ul id="footer-info"> <li id="footer-info-lastmod"> Sidan redigerades senast den 14 mars 2021 kl. 18.23.</li> <li id="footer-info-copyright">Wikipedias text är tillgänglig under licensen <a rel="nofollow" class="external text" href="//creativecommons.org/licenses/by-sa/4.0/deed.sv">Creative Commons Erkännande-dela-lika 4.0 Unported</a>. 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Heltalspartition – Wikipedia