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class="vector-pinned-container"> <div id="vector-toc" class="vector-toc vector-pinnable-element"> <div class="vector-pinnable-header vector-toc-pinnable-header vector-pinnable-header-pinned" data-feature-name="toc-pinned" data-pinnable-element-id="vector-toc" > <h2 class="vector-pinnable-header-label">Cuprins</h2> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-toc.pin">mută în bara laterală</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">ascunde</button> </div> <ul class="vector-toc-contents" id="mw-panel-toc-list"> <li id="toc-mw-content-text" class="vector-toc-list-item vector-toc-level-1"> <a href="#" class="vector-toc-link"> <div class="vector-toc-text">Început</div> </a> </li> <li id="toc-Exemple" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Exemple"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>Exemple</span> </div> </a> <ul id="toc-Exemple-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Construcții" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Construcții"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Construcții</span> </div> </a> <ul id="toc-Construcții-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Secvențe_asociate" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Secvențe_asociate"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Secvențe asociate</span> </div> </a> <ul id="toc-Secvențe_asociate-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Relația_de_recurență" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Relația_de_recurență"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Relația de recurență</span> </div> </a> <ul id="toc-Relația_de_recurență-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Funcția_generatoare" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Funcția_generatoare"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Funcția generatoare</span> </div> </a> <ul id="toc-Funcția_generatoare-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Utilizări" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Utilizări"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Utilizări</span> </div> </a> <ul id="toc-Utilizări-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Note" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Note"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Note</span> </div> </a> <ul id="toc-Note-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Lectură_suplimentară" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Lectură_suplimentară"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>Lectură suplimentară</span> </div> </a> <ul id="toc-Lectură_suplimentară-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Vezi_și" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Vezi_și"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>Vezi și</span> </div> </a> <ul id="toc-Vezi_și-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Legături_externe" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Legături_externe"> <div class="vector-toc-text"> <span class="vector-toc-numb">10</span> <span>Legături externe</span> </div> </a> <ul id="toc-Legături_externe-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="Cuprins" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <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="Comută cuprinsul" > <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">Comută cuprinsul</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">Număr Schröder</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="Mergeți la un articol în altă limbă. Disponibil în 8 limbi" > <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-8" 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">8 limbi</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%B9%D8%AF%D8%AF_%D8%B4%D8%B1%D9%88%D8%AF%D8%B1" title="عدد شرودر – arabă" lang="ar" hreflang="ar" data-title="عدد شرودر" data-language-autonym="العربية" data-language-local-name="arabă" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Schr%C3%B6der-Zahlen" title="Schröder-Zahlen – germană" lang="de" hreflang="de" data-title="Schröder-Zahlen" data-language-autonym="Deutsch" data-language-local-name="germană" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Schr%C3%B6der_number" title="Schröder number – engleză" lang="en" hreflang="en" data-title="Schröder number" data-language-autonym="English" data-language-local-name="engleză" class="interlanguage-link-target"><span>English</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Nombre_de_Schr%C3%B6der" title="Nombre de Schröder – franceză" lang="fr" hreflang="fr" data-title="Nombre de Schröder" data-language-autonym="Français" data-language-local-name="franceză" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Numero_di_Schr%C3%B6der" title="Numero di Schröder – italiană" lang="it" hreflang="it" data-title="Numero di Schröder" data-language-autonym="Italiano" data-language-local-name="italiană" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%A7%D0%B8%D1%81%D0%BB%D0%B0_%D0%A8%D1%80%D1%91%D0%B4%D0%B5%D1%80%D0%B0" title="Числа Шрёдера – rusă" lang="ru" hreflang="ru" data-title="Числа Шрёдера" data-language-autonym="Русский" data-language-local-name="rusă" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Schr%C3%B6derjevo_%C5%A1tevilo" title="Schröderjevo število – slovenă" lang="sl" hreflang="sl" data-title="Schröderjevo število" data-language-autonym="Slovenščina" data-language-local-name="slovenă" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Schr%C3%B6dertal" title="Schrödertal – suedeză" lang="sv" hreflang="sv" data-title="Schrödertal" data-language-autonym="Svenska" data-language-local-name="suedeză" class="interlanguage-link-target"><span>Svenska</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/Q2071909#sitelinks-wikipedia" title="Modifică legăturile interlinguale" class="wbc-editpage">Modifică legăturile</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="Spații de nume"> <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/Num%C4%83r_Schr%C3%B6der" title="Vedeți conținutul paginii [a]" accesskey="a"><span>Articol</span></a></li><li id="ca-talk" class="vector-tab-noicon mw-list-item"><a href="/wiki/Discu%C8%9Bie:Num%C4%83r_Schr%C3%B6der" rel="discussion" title="Discuții despre această pagină [t]" accesskey="t"><span>Discuție</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="Change language variant" > <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">română</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="Vizualizări"> <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/Num%C4%83r_Schr%C3%B6der"><span>Lectură</span></a></li><li id="ca-ve-edit" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=Num%C4%83r_Schr%C3%B6der&veaction=edit" title="Modificați această pagină cu EditorulVizual [v]" accesskey="v"><span>Modificare</span></a></li><li id="ca-edit" class="collapsible vector-tab-noicon mw-list-item"><a href="/w/index.php?title=Num%C4%83r_Schr%C3%B6der&action=edit" title="Modificați codul sursă al acestei pagini [e]" accesskey="e"><span>Modificare sursă</span></a></li><li id="ca-history" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=Num%C4%83r_Schr%C3%B6der&action=history" title="Versiunile anterioare ale paginii și autorii lor. [h]" accesskey="h"><span>Istoric</span></a></li> </ul> </div> </div> </nav> <nav class="vector-page-tools-landmark" aria-label="Page tools"> <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="Unelte" > <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">Unelte</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">Unelte</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-page-tools.pin">mută în bara laterală</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-page-tools.unpin">ascunde</button> </div> <div id="p-cactions" class="vector-menu mw-portlet mw-portlet-cactions emptyPortlet vector-has-collapsible-items" title="Mai multe opțiuni" > <div class="vector-menu-heading"> Acțiuni </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/Num%C4%83r_Schr%C3%B6der"><span>Lectură</span></a></li><li id="ca-more-ve-edit" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=Num%C4%83r_Schr%C3%B6der&veaction=edit" title="Modificați această pagină cu EditorulVizual [v]" accesskey="v"><span>Modificare</span></a></li><li id="ca-more-edit" class="collapsible vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=Num%C4%83r_Schr%C3%B6der&action=edit" title="Modificați codul sursă al acestei pagini [e]" accesskey="e"><span>Modificare sursă</span></a></li><li id="ca-more-history" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=Num%C4%83r_Schr%C3%B6der&action=history"><span>Istoric</span></a></li> </ul> </div> </div> <div id="p-tb" class="vector-menu mw-portlet mw-portlet-tb" > <div class="vector-menu-heading"> General </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="t-whatlinkshere" class="mw-list-item"><a href="/wiki/Special:Ce_se_leag%C4%83_aici/Num%C4%83r_Schr%C3%B6der" title="Lista tuturor paginilor wiki care conduc spre această pagină [j]" accesskey="j"><span>Ce trimite aici</span></a></li><li id="t-recentchangeslinked" class="mw-list-item"><a href="/wiki/Special:Modific%C4%83ri_corelate/Num%C4%83r_Schr%C3%B6der" rel="nofollow" title="Schimbări recente în legătură cu această pagină [k]" accesskey="k"><span>Schimbări corelate</span></a></li><li id="t-upload" class="mw-list-item"><a href="/wiki/Wikipedia:Trimite_fi%C8%99ier" title="Încărcare fișiere [u]" accesskey="u"><span>Trimite fișier</span></a></li><li id="t-specialpages" class="mw-list-item"><a href="/wiki/Special:Pagini_speciale" title="Lista tuturor paginilor speciale [q]" accesskey="q"><span>Pagini speciale</span></a></li><li id="t-permalink" class="mw-list-item"><a href="/w/index.php?title=Num%C4%83r_Schr%C3%B6der&oldid=15806940" title="Legătură permanentă către această versiune a acestei pagini"><span>Legătură permanentă</span></a></li><li id="t-info" class="mw-list-item"><a href="/w/index.php?title=Num%C4%83r_Schr%C3%B6der&action=info" title="Mai multe informații despre această pagină"><span>Informații despre pagină</span></a></li><li id="t-cite" class="mw-list-item"><a href="/w/index.php?title=Special:Citeaz%C4%83&page=Num%C4%83r_Schr%C3%B6der&id=15806940&wpFormIdentifier=titleform" title="Informații cu privire la modul de citare a acestei pagini"><span>Citează acest articol</span></a></li><li id="t-urlshortener" class="mw-list-item"><a href="/w/index.php?title=Special:UrlShortener&url=https%3A%2F%2Fro.wikipedia.org%2Fwiki%2FNum%25C4%2583r_Schr%25C3%25B6der"><span>Obține URL scurtat</span></a></li><li id="t-urlshortener-qrcode" class="mw-list-item"><a href="/w/index.php?title=Special:QrCode&url=https%3A%2F%2Fro.wikipedia.org%2Fwiki%2FNum%25C4%2583r_Schr%25C3%25B6der"><span>Descărcați codul QR</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"> Tipărire/exportare </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:Carte&bookcmd=book_creator&referer=Num%C4%83r+Schr%C3%B6der"><span>Creare carte</span></a></li><li id="coll-download-as-rl" class="mw-list-item"><a href="/w/index.php?title=Special:DownloadAsPdf&page=Num%C4%83r_Schr%C3%B6der&action=show-download-screen"><span>Descărcare ca PDF</span></a></li><li id="t-print" class="mw-list-item"><a href="/w/index.php?title=Num%C4%83r_Schr%C3%B6der&printable=yes" title="Versiunea de tipărit a acestei pagini [p]" accesskey="p"><span>Versiune de tipărit</span></a></li> </ul> </div> </div> <div id="p-wikibase-otherprojects" class="vector-menu mw-portlet mw-portlet-wikibase-otherprojects" > <div class="vector-menu-heading"> În alte proiecte </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="wb-otherproject-link wb-otherproject-commons mw-list-item"><a href="https://commons.wikimedia.org/wiki/Category:Schr%C3%B6der_number" hreflang="en"><span>Wikimedia Commons</span></a></li><li id="t-wikibase" class="wb-otherproject-link wb-otherproject-wikibase-dataitem mw-list-item"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q2071909" title="Legătură către elementul asociat din depozitul de date [g]" accesskey="g"><span>Element Wikidata</span></a></li> </ul> </div> </div> </div> </div> </div> </div> </nav> </div> </div> </div> <div class="vector-column-end"> <div class="vector-sticky-pinned-container"> <nav class="vector-page-tools-landmark" aria-label="Page tools"> <div id="vector-page-tools-pinned-container" class="vector-pinned-container"> </div> </nav> <nav class="vector-appearance-landmark" aria-label="Aspect"> <div id="vector-appearance-pinned-container" class="vector-pinned-container"> <div id="vector-appearance" class="vector-appearance vector-pinnable-element"> <div class="vector-pinnable-header vector-appearance-pinnable-header vector-pinnable-header-pinned" data-feature-name="appearance-pinned" data-pinnable-element-id="vector-appearance" data-pinned-container-id="vector-appearance-pinned-container" data-unpinned-container-id="vector-appearance-unpinned-container" > <div class="vector-pinnable-header-label">Aspect</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-appearance.pin">mută în bara laterală</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-appearance.unpin">ascunde</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">De la Wikipedia, enciclopedia liberă</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="ro" dir="ltr"><table class="infobox vcard" cellspacing="5" style="width: 22em; text-align: left; font-size: 88%; line-height: 1.5em;"><caption class="fn org" style="font-size: 125%; font-weight: bold;">Număr Schröder</caption> <tbody><tr><th style="">Numit după</th><td class="" style="">Ernst Schröder</td></tr><tr><th style=""><abbr title="Număr">Nr.</abbr> de termeni cunoscuți</th><td class="" style=""><a href="/wiki/Infinit" title="Infinit">infinit</a></td></tr><tr><th style="">Primii termeni</th><td class="" style=""><a href="/wiki/1_(cifr%C4%83)" title="1 (cifră)">1</a>, <a href="/wiki/2_(cifr%C4%83)" title="2 (cifră)">2</a>, <a href="/wiki/6_(cifr%C4%83)" title="6 (cifră)">6</a>, <a href="/wiki/22_(num%C4%83r)" title="22 (număr)">22</a>, <a href="/wiki/90_(num%C4%83r)" title="90 (număr)">90</a>, 394, 1806</td></tr><tr><th style="">Index <a href="/wiki/Enciclopedia_electronic%C4%83_a_%C8%99irurilor_de_numere_%C3%AEntregi" title="Enciclopedia electronică a șirurilor de numere întregi">OEIS</a></th><td class="" style=""><ul style="list-style:none none; padding:0px; margin:0px"><li><a rel="nofollow" class="external text" href="//oeis.org/A006318">A006318</a></li><li>Large Schröder</li></ul></td></tr> </tbody></table> <p>În <a href="/wiki/Matematic%C4%83" title="Matematică">matematică</a>, un <b>număr Schröder</b> <span class="mwe-math-element"><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> <mo>,</mo> </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/a944e563075186a3887013295fdbeb922903210e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.29ex; height:2.509ex;" alt="{\displaystyle S_{n},}"></span> numit și <i>număr Schröder mare</i><sup id="cite_ref-C77_1-0" class="reference"><a href="#cite_note-C77-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup>, descrie numărul de căi dintr-o grilă de la colțul de sud-vest <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (0,0)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (0,0)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5d630d3e781a53b0a3559ae7e5b45f9479a3141a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.168ex; height:2.843ex;" alt="{\displaystyle (0,0)}"></span> al grilei <span class="mwe-math-element"><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\times n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>×</mo> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\times n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59d2b4cb72e304526cf5b5887147729ea259da78" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.63ex; height:1.676ex;" alt="{\displaystyle n\times n}"></span> până la colțul de nord-est <span class="mwe-math-element"><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,n),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>n</mi> <mo>,</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (n,n),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5cb00b991b45039cfcf97772e3be2a4114893d9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.28ex; height:2.843ex;" alt="{\displaystyle (n,n),}"></span> folosind doar pași simpli spre nord, <span class="mwe-math-element"><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 stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo 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/c79c6838e423c1ed3c7ea532a56dc9f9dae8290b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.168ex; height:2.843ex;" alt="{\displaystyle (0,1)}"></span>, nord-est, <span class="mwe-math-element"><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)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (1,1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2c2a42feb07f4139bf871ae6856b11d4567bea23" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.168ex; height:2.843ex;" alt="{\displaystyle (1,1)}"></span> sau spre est, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (1,0)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (1,0)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5b53cc1773694affcc1d4d6c2c778d43156a1206" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.168ex; height:2.843ex;" alt="{\displaystyle (1,0)}"></span> care nu se ridică deasupra diagonalei SW–NE.<sup id="cite_ref-A006318_2-0" class="reference"><a href="#cite_note-A006318-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p><p>Primele câteva numere Schröder sunt:<sup id="cite_ref-C77_1-1" class="reference"><a href="#cite_note-C77-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-A006318_2-1" class="reference"><a href="#cite_note-A006318-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><a href="/wiki/1_(cifr%C4%83)" title="1 (cifră)">1</a>, <a href="/wiki/2_(cifr%C4%83)" title="2 (cifră)">2</a>, <a href="/wiki/6_(cifr%C4%83)" title="6 (cifră)">6</a>, <a href="/wiki/22_(num%C4%83r)" title="22 (număr)">22</a>, <a href="/wiki/90_(num%C4%83r)" title="90 (număr)">90</a>, 394, 1806, 8558, 41586, 206098, 1037718, 5293446, 27297738, 142078746, 745387038, 3937603038, 20927156706, 111818026018, 600318853926, 3236724317174, 17518619320890, 95149655201962, 518431875418926, 2832923350929742, 15521467648875090, ...</dd></dl> <p>Ele au fost numite după <a href="/wiki/Matematician" title="Matematician">matematicianul</a> <a href="/wiki/German" class="mw-redirect" title="German">german</a> <a href="/w/index.php?title=Ernst_Schr%C3%B6der&action=edit&redlink=1" class="new" title="Ernst Schröder — pagină inexistentă">Ernst Schröder</a>. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Exemple">Exemple</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Num%C4%83r_Schr%C3%B6der&veaction=edit&section=1" title="Modifică secțiunea: Exemple" class="mw-editsection-visualeditor"><span>modificare</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Num%C4%83r_Schr%C3%B6der&action=edit&section=1" title="Edit section's source code: Exemple"><span>modificare sursă</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Următoarea figură arată cele 6 astfel de căi într-o grilă <span class="mwe-math-element"><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\times 2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mo>×</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2\times 2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f8a0e3400ffb97d67c00267ed50cddfe824cbe80" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.165ex; height:2.176ex;" alt="{\displaystyle 2\times 2}"></span>: <span class="mw-default-size" typeof="mw:File"><a href="/wiki/Fi%C8%99ier:Schroeder_number_2x2.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e6/Schroeder_number_2x2.svg/600px-Schroeder_number_2x2.svg.png" decoding="async" width="600" height="100" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/e6/Schroeder_number_2x2.svg/900px-Schroeder_number_2x2.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/e6/Schroeder_number_2x2.svg/1200px-Schroeder_number_2x2.svg.png 2x" data-file-width="600" data-file-height="100" /></a></span> </p> <div class="mw-heading mw-heading2"><h2 id="Construcții"><span id="Construc.C8.9Bii"></span>Construcții</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Num%C4%83r_Schr%C3%B6der&veaction=edit&section=2" title="Modifică secțiunea: Construcții" class="mw-editsection-visualeditor"><span>modificare</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Num%C4%83r_Schr%C3%B6der&action=edit&section=2" title="Edit section's source code: Construcții"><span>modificare sursă</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>O cale Schröder de lungime <span class="mwe-math-element"><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> este o cale în grilă de la <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (0,0)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (0,0)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5d630d3e781a53b0a3559ae7e5b45f9479a3141a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.168ex; height:2.843ex;" alt="{\displaystyle (0,0)}"></span> la <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (2n,0)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>2</mn> <mi>n</mi> <mo>,</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (2n,0)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/167732e6bccde018de275f31386b4613255ebfd7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.563ex; height:2.843ex;" alt="{\displaystyle (2n,0)}"></span> cu pași spre nord-est, <span class="mwe-math-element"><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),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (1,1),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/06826b80ffeef4bd73b0243f7ccab42357da621e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.815ex; height:2.843ex;" alt="{\displaystyle (1,1),}"></span> est, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (2,0)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>2</mn> <mo>,</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (2,0)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/03972a1096453eb69d2337a01165887760403a8f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.168ex; height:2.843ex;" alt="{\displaystyle (2,0)}"></span> și sud-est, <span class="mwe-math-element"><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),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (1,-1),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fb6970fc51a5b6f87156198f776b07be18a3a4ad" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.623ex; height:2.843ex;" alt="{\displaystyle (1,-1),}"></span> care nu coboară sub axa <span class="mwe-math-element"><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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span>. Al <span class="texhtml mvar" style="font-style:italic;">n</span>-lea număr Schröder este numărul căilor Schröder de lungime <span class="mwe-math-element"><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>.<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> Următoarea figură arată cele 6 căi Schröder de lungime 2: <span class="mw-default-size" typeof="mw:File"><a href="/wiki/Fi%C8%99ier:Schroeder_paths.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/37/Schroeder_paths.svg/610px-Schroeder_paths.svg.png" decoding="async" width="610" height="80" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/37/Schroeder_paths.svg/915px-Schroeder_paths.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/37/Schroeder_paths.svg/1220px-Schroeder_paths.svg.png 2x" data-file-width="610" data-file-height="80" /></a></span> </p><p>Similar, numerele Schröder enumeră modalitățile de a împărți un <a href="/wiki/Dreptunghi" title="Dreptunghi">dreptunghi</a> în <span class="mwe-math-element"><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+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n+1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2a135e65a42f2d73cccbfc4569523996ca0036f1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.398ex; height:2.343ex;" alt="{\displaystyle n+1}"></span> dreptunghiuri mai mici folosind <span class="mwe-math-element"><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> tăieturi prin <span class="mwe-math-element"><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> puncte oarecare date în interiorul dreptunghiului, fiecare tăietură intersectând unul dintre puncte și împărțind doar un singur dreptunghi în două (adică, numărul de structuri de tip <a href="/w/index.php?title=Parti%C8%9Bie_ghilotin%C4%83&action=edit&redlink=1" class="new" title="Partiție ghilotină — pagină inexistentă">partiții ghilotină</a> diferite). Acest lucru este similar cu procesul de <a href="/wiki/Triangularea_unui_poligon" title="Triangularea unui poligon">triangulare</a>, în care o formă este împărțită în triunghiuri care nu se suprapun în loc de dreptunghiuri. Următoarea figură arată cele 6 astfel de împărțiri ale unui dreptunghi în 3 dreptunghiuri folosind două tăieturi: <span class="mw-default-size" typeof="mw:File"><a href="/wiki/Fi%C8%99ier:Schroeder_rectangulation_3.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/d2/Schroeder_rectangulation_3.svg/600px-Schroeder_rectangulation_3.svg.png" decoding="async" width="600" height="100" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/d2/Schroeder_rectangulation_3.svg/900px-Schroeder_rectangulation_3.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/d2/Schroeder_rectangulation_3.svg/1200px-Schroeder_rectangulation_3.svg.png 2x" data-file-width="600" data-file-height="100" /></a></span> </p><p>În imaginea de mai jos sunt cele 22 de împărțiri ale unui dreptunghi în 4 dreptunghiuri folosind trei tăieturi: <span class="mw-default-size" typeof="mw:File"><a href="/wiki/Fi%C8%99ier:Schroeder_rectangulation_4.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/5d/Schroeder_rectangulation_4.svg/555px-Schroeder_rectangulation_4.svg.png" decoding="async" width="555" height="100" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/5d/Schroeder_rectangulation_4.svg/833px-Schroeder_rectangulation_4.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/5d/Schroeder_rectangulation_4.svg/1110px-Schroeder_rectangulation_4.svg.png 2x" data-file-width="555" data-file-height="100" /></a></span> </p><p>Numărul Schröder <span class="mwe-math-element"><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> indică și <a href="/w/index.php?title=Permutare_separabil%C4%83&action=edit&redlink=1" class="new" title="Permutare separabilă — pagină inexistentă">permutările separabile</a> de lungime <span class="mwe-math-element"><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-1.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>−</mo> <mn>1.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n-1.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6abe7e8ef775e730e29e170abf3f83a604df2ec6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.044ex; height:2.343ex;" alt="{\displaystyle n-1.}"></span> </p> <div class="mw-heading mw-heading2"><h2 id="Secvențe_asociate"><span id="Secven.C8.9Be_asociate"></span>Secvențe asociate</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Num%C4%83r_Schr%C3%B6der&veaction=edit&section=3" title="Modifică secțiunea: Secvențe asociate" class="mw-editsection-visualeditor"><span>modificare</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Num%C4%83r_Schr%C3%B6der&action=edit&section=3" title="Edit section's source code: Secvențe asociate"><span>modificare sursă</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Numerele Schröder sunt numite uneori numere Schröder „mari” deoarece există o altă succesiune Schröder: „numerele Schröder mici”, cunoscute și sub numele <a href="/wiki/Num%C4%83r_Schr%C3%B6der%E2%80%93Hiparh" title="Număr Schröder–Hiparh">numere Schröder–Hiparh</a> sau „ numere super Catalan”. Conexiunile dintre aceste căi pot fi văzute în câteva moduri: </p> <ul><li>Fie căile de la <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (0,0)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (0,0)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5d630d3e781a53b0a3559ae7e5b45f9479a3141a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.168ex; height:2.843ex;" alt="{\displaystyle (0,0)}"></span> la <span class="mwe-math-element"><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,n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>n</mi> <mo>,</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (n,n)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af3b144aa2936855ba778dcc458ae30b5d119924" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.633ex; height:2.843ex;" alt="{\displaystyle (n,n)}"></span> cu pașii <span class="mwe-math-element"><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),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (1,1),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/06826b80ffeef4bd73b0243f7ccab42357da621e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.815ex; height:2.843ex;" alt="{\displaystyle (1,1),}"></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 (2,0)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>2</mn> <mo>,</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (2,0)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/03972a1096453eb69d2337a01165887760403a8f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.168ex; height:2.843ex;" alt="{\displaystyle (2,0)}"></span> și <span class="mwe-math-element"><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)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (1,-1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/731b59b333c7e26d441f64aa67454eaf2ce21412" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.976ex; height:2.843ex;" alt="{\displaystyle (1,-1)}"></span> care nu se ridică deasupra diagonalei principale. Există două tipuri de căi: cele care au mișcări de-a lungul diagonalei principale și cele care nu. Numerele Schröder (mari) enumeră ambele tipuri de căi, iar numerele Schröder mici enumeră doar căile care ating diagonala, dar nu au pași de-a lungul ei.<sup id="cite_ref-A001003_4-0" class="reference"><a href="#cite_note-A001003-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup></li></ul> <ul><li>Așa cum există căi Schröder (mari), o cale Schröder mică este o cale Schröder care nu are trepte orizontale pe axa <span class="mwe-math-element"><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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span>.<sup id="cite_ref-Bijections_with_Dyck_paths_5-0" class="reference"><a href="#cite_note-Bijections_with_Dyck_paths-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup></li></ul> <ul><li>Dacă <span class="mwe-math-element"><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> este al <span class="texhtml mvar" style="font-style:italic;">n</span>-lea număr Schröder și <span class="mwe-math-element"><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/d671890050b21484dde3087d000700c97fc3b03c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.309ex; height:2.009ex;" alt="{\displaystyle s_{n}}"></span> este al <span class="texhtml mvar" style="font-style:italic;">n</span>-lea număr Schröder mic, atunci <span class="mwe-math-element"><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}=2s_{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> <mo>=</mo> <mn>2</mn> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S_{n}=2s_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/25a3a1327d3465b02b308c5d26cf9ce8f74b05df" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.213ex; height:2.509ex;" alt="{\displaystyle S_{n}=2s_{n}}"></span> pentru <span class="mwe-math-element"><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>0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n>0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/27a6a5d982d54202a14f111cb8a49210501b2c96" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.656ex; height:2.176ex;" alt="{\displaystyle n>0}"></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 (S_{0}=s_{0}=1).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>=</mo> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>=</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (S_{0}=s_{0}=1).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b9b885dbbb00e36b3847d2445e516a09d7f6e115" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.439ex; height:2.843ex;" alt="{\displaystyle (S_{0}=s_{0}=1).}"></span><sup id="cite_ref-Bijections_with_Dyck_paths_5-1" class="reference"><a href="#cite_note-Bijections_with_Dyck_paths-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup></li></ul> <p>Căile Schröder sunt similare cu căile Dyck, dar admit și pasul orizontal în loc de doar pașii diagonali. Alte căi similară sunt căile <a href="/wiki/Num%C4%83r_Motzkin" title="Număr Motzkin">Motzkin</a>, care sunt aceleași căi diagonale, dar admit doar un singur pas orizontal, (1,0), și enumeră astfel de căi de la <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (0,0)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (0,0)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5d630d3e781a53b0a3559ae7e5b45f9479a3141a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.168ex; height:2.843ex;" alt="{\displaystyle (0,0)}"></span> la <span class="mwe-math-element"><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,0)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>n</mi> <mo>,</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (n,0)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5fd8a7c3a302914ba5ae7cac4d8df11b59943934" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.4ex; height:2.843ex;" alt="{\displaystyle (n,0)}"></span>.<sup id="cite_ref-Catalan,_Motzkin,_and_Schröder_numbers_6-0" class="reference"><a href="#cite_note-Catalan,_Motzkin,_and_Schröder_numbers-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> </p><p>Există și o <a href="/w/index.php?title=Matrice_triunghiular%C4%83&action=edit&redlink=1" class="new" title="Matrice triunghiulară — pagină inexistentă">matrice triunghiulară</a> asociată cu numerele Schröder, care dă o <a href="/wiki/Rela%C8%9Bie_de_recuren%C8%9B%C4%83" title="Relație de recurență">relație de recurență</a><sup id="cite_ref-A033877_7-0" class="reference"><a href="#cite_note-A033877-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup>(însă nu doar între numerele Schröder). Primii termeni din matrice sunt:<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> </p> <dl><dd>1, 1, 2, 1, 4, 6, 1, 6, 16, 22, ...</dd></dl> <p>Este mai ușor de văzut conexiunea cu numerele Schröder atunci când secvența este în forma sa triunghiulară: </p> <table class="wikitable" style="text-align:center;"> <tbody><tr> <th><br />    <span class="texhtml mvar" style="font-style:italic;">m</span><br /><span class="texhtml mvar" style="font-style:italic;">n</span> </th> <th width="50">0 </th> <th width="50">1 </th> <th width="50">2 </th> <th width="50">3 </th> <th width="50">4 </th> <th width="50">5 </th> <th width="50">6 </th></tr> <tr> <th>0 </th> <td>1</td> <td></td> <td></td> <td></td> <td></td> <td></td> <td> </td></tr> <tr> <th>1 </th> <td>1</td> <td>2</td> <td></td> <td></td> <td></td> <td></td> <td> </td></tr> <tr> <th>2 </th> <td>1</td> <td>4</td> <td>6</td> <td></td> <td></td> <td></td> <td> </td></tr> <tr> <th>3 </th> <td>1</td> <td>6</td> <td>16</td> <td>22</td> <td></td> <td></td> <td> </td></tr> <tr> <th>4 </th> <td>1</td> <td>8</td> <td>30</td> <td>68</td> <td>90</td> <td></td> <td> </td></tr> <tr> <th>5 </th> <td>1</td> <td>10</td> <td>48</td> <td>146</td> <td>304</td> <td>394</td> <td> </td></tr> <tr> <th>6 </th> <td>1</td> <td>12</td> <td>70</td> <td>264</td> <td>714</td> <td>1412</td> <td>1806 </td></tr></tbody></table> <p>Aici numerele Schröder apar pe diagonală, de exemplu <span class="mwe-math-element"><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}=T(n,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> <mo>=</mo> <mi>T</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>,</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S_{n}=T(n,n)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/598d2e0b030dc8bf4074bda85e3e53f3477dbb3f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.011ex; height:2.843ex;" alt="{\displaystyle S_{n}=T(n,n)}"></span> unde <span class="mwe-math-element"><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(n,k)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</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 T(n,k)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/392e446e9de490a7ef9cc8955f693a8ae2ed34f5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.085ex; height:2.843ex;" alt="{\displaystyle T(n,k)}"></span> este <a href="/wiki/Valoare_(matematic%C4%83)" title="Valoare (matematică)">valoarea</a> din rândul <span class="texhtml mvar" style="font-style:italic;">n</span> și coloana <span class="texhtml mvar" style="font-style:italic;">k</span>. Relația de recurență dată de această aranjare este </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 T(n,k)=T(n,k-1)+T(n-1,k-1)+T(n-1,k)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>,</mo> <mi>k</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>T</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>,</mo> <mi>k</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>+</mo> <mi>T</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> <mo>+</mo> <mi>T</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo>,</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T(n,k)=T(n,k-1)+T(n-1,k-1)+T(n-1,k)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/955012f78b394d82370bc2bb86f21267782d3390" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:53.132ex; height:2.843ex;" alt="{\displaystyle T(n,k)=T(n,k-1)+T(n-1,k-1)+T(n-1,k)}"></span></dd></dl> <p>cu <span class="mwe-math-element"><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,k)=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mi>k</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T(1,k)=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0060ae193d5a71f7f35d51c7c94d751225133fae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.114ex; height:2.843ex;" alt="{\displaystyle T(1,k)=1}"></span> și <span class="mwe-math-element"><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(n,k)=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>,</mo> <mi>k</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T(n,k)=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/df3ddeae79350a815de7b362bdfd5c34343c4d25" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.346ex; height:2.843ex;" alt="{\displaystyle T(n,k)=0}"></span> pentru <span class="mwe-math-element"><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/66e81682bf174c978e9008ffb557ba4da2cf7478" 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>.<sup id="cite_ref-A033877_7-1" class="reference"><a href="#cite_note-A033877-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> O altă observație interesantă este că suma din cel de al <span class="texhtml mvar" style="font-style:italic;">n</span>-lea rând este al (<span class="texhtml mvar" style="font-style:italic;">n</span>+1)-lea număr Schröder mic, adică </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}^{n}T(n,k)=s_{n+1}}"> <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>n</mi> </mrow> </munderover> <mi>T</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>,</mo> <mi>k</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{k=0}^{n}T(n,k)=s_{n+1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/73588ec34b3dd87a865db5cd74a142768cd91104" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:18.335ex; height:7.009ex;" alt="{\displaystyle \sum _{k=0}^{n}T(n,k)=s_{n+1}}"></span>.</dd></dl> <div class="mw-heading mw-heading2"><h2 id="Relația_de_recurență"><span id="Rela.C8.9Bia_de_recuren.C8.9B.C4.83"></span>Relația de recurență</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Num%C4%83r_Schr%C3%B6der&veaction=edit&section=4" title="Modifică secțiunea: Relația de recurență" class="mw-editsection-visualeditor"><span>modificare</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Num%C4%83r_Schr%C3%B6der&action=edit&section=4" title="Edit section's source code: Relația de recurență"><span>modificare sursă</span></a><span class="mw-editsection-bracket">]</span></span></div> <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_{n}=3S_{n-1}+\sum _{k=1}^{n-2}S_{k}S_{n-k-1}}"> <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> <mo>=</mo> <mn>3</mn> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <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>n</mi> <mo>−</mo> <mn>2</mn> </mrow> </munderover> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mi>k</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S_{n}=3S_{n-1}+\sum _{k=1}^{n-2}S_{k}S_{n-k-1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4c4f2626bbe1553989826dc10c8e0df89fc0ca53" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:27.623ex; height:7.343ex;" alt="{\displaystyle S_{n}=3S_{n-1}+\sum _{k=1}^{n-2}S_{k}S_{n-k-1}}"></span> pentru <span class="mwe-math-element"><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> cu <span class="mwe-math-element"><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_{0}=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S_{0}=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/261e25d4953392baa666e1c5f3dec3db1bbd94f4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.74ex; height:2.509ex;" alt="{\displaystyle S_{0}=1}"></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 S_{1}=2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S_{1}=2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/40f6c0d533650abb1116fd44a4e13108404faf06" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.74ex; height:2.509ex;" alt="{\displaystyle S_{1}=2}"></span>.<sup id="cite_ref-Schröder_number_formulas_9-0" class="reference"><a href="#cite_note-Schröder_number_formulas-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Funcția_generatoare"><span id="Func.C8.9Bia_generatoare"></span>Funcția generatoare</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Num%C4%83r_Schr%C3%B6der&veaction=edit&section=5" title="Modifică secțiunea: Funcția generatoare" class="mw-editsection-visualeditor"><span>modificare</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Num%C4%83r_Schr%C3%B6der&action=edit&section=5" title="Edit section's source code: Funcția generatoare"><span>modificare sursă</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/w/index.php?title=Func%C8%9Bie_generatoare&action=edit&redlink=1" class="new" title="Funcție generatoare — pagină inexistentă">Funcția generatoare</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(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1d6d96c680c58289ec8857273d6938cacd742084" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.966ex; height:2.843ex;" alt="{\displaystyle G(x)}"></span> 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"> <mo stretchy="false">(</mo> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> </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/fbaf06abcc2edbf3daef7fd2bb36b75952ffe035" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.453ex; height:2.843ex;" alt="{\displaystyle (S_{n})}"></span> este<sup id="cite_ref-Schröder_number_formulas_9-1" class="reference"><a href="#cite_note-Schröder_number_formulas-9"><span class="cite-bracket">[</span>9<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 G(x)={\frac {1-x-{\sqrt {x^{2}-6x+1}}}{2x}}=\sum _{n=0}^{\infty }S_{n}x^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>1</mn> <mo>−</mo> <mi>x</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>6</mn> <mi>x</mi> <mo>+</mo> <mn>1</mn> </msqrt> </mrow> </mrow> <mrow> <mn>2</mn> <mi>x</mi> </mrow> </mfrac> </mrow> <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> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G(x)={\frac {1-x-{\sqrt {x^{2}-6x+1}}}{2x}}=\sum _{n=0}^{\infty }S_{n}x^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e908c1ff8e74b47f2505d19d9f97313c5156e785" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:43.148ex; height:7.343ex;" alt="{\displaystyle G(x)={\frac {1-x-{\sqrt {x^{2}-6x+1}}}{2x}}=\sum _{n=0}^{\infty }S_{n}x^{n}}"></span>.</dd></dl> <div class="mw-heading mw-heading2"><h2 id="Utilizări"><span id="Utiliz.C4.83ri"></span>Utilizări</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Num%C4%83r_Schr%C3%B6der&veaction=edit&section=6" title="Modifică secțiunea: Utilizări" class="mw-editsection-visualeditor"><span>modificare</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Num%C4%83r_Schr%C3%B6der&action=edit&section=6" title="Edit section's source code: Utilizări"><span>modificare sursă</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Fi%C8%99ier:Diamant_azteque_plein.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/2/20/Diamant_azteque_plein.svg/220px-Diamant_azteque_plein.svg.png" decoding="async" width="220" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/20/Diamant_azteque_plein.svg/330px-Diamant_azteque_plein.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/20/Diamant_azteque_plein.svg/440px-Diamant_azteque_plein.svg.png 2x" data-file-width="323" data-file-height="323" /></a><figcaption>Diamant aztec de ordinul 4</figcaption></figure> <p>Un subiect al combinatoricii sunt formele <a href="/wiki/Pavare" title="Pavare">pavărilor</a>, iar un exemplu particular al acestora este <a href="/w/index.php?title=Pavare_cu_dominouri&action=edit&redlink=1" class="new" title="Pavare cu dominouri — pagină inexistentă">pavarea cu dominouri</a>. În acest caz întrebarea este: „câte dominouri (adică câte dreptunghiuri <span class="mwe-math-element"><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\times 2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>×</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1\times 2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1797defbfda89e7b60e17a72380578326618bccb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.165ex; height:2.176ex;" alt="{\displaystyle 1\times 2}"></span> sau <span class="mwe-math-element"><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\times 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mo>×</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2\times 1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9206e2cc500de17a162b111d3c2cf61ec68c97a8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.165ex; height:2.176ex;" alt="{\displaystyle 2\times 1}"></span>) se pot aranja într-o formă astfel încât niciuna dintre dominouri să nu se suprapună, întreaga formă să fie acoperită și niciunul dintre dominouri să nu iasă afară din formă?". Forma cu care au legătură numerele Schröder este <a href="/w/index.php?title=Diamant_aztec&action=edit&redlink=1" class="new" title="Diamant aztec — pagină inexistentă">diamantul aztec</a>. Alături este prezentat un diamant aztec de ordinul 4 cu o posibilă pavare domino. </p><p>Se pare că <a href="/wiki/Determinant_(matematic%C4%83)" title="Determinant (matematică)">determinantul</a> unei <a href="/w/index.php?title=Matrice_Hankel&action=edit&redlink=1" class="new" title="Matrice Hankel — pagină inexistentă">matrice Hankel</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 (2n-1)\times (2n-1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>2</mn> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>×</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (2n-1)\times (2n-1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4bd9008e9bc8e7d36191ada6b91432af0f5dae7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.579ex; height:2.843ex;" alt="{\displaystyle (2n-1)\times (2n-1)}"></span> de numere Schröder, adică matricea pătrată al cărei element <span class="mwe-math-element"><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,j)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>i</mi> <mo>,</mo> <mi>j</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (i,j)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8ef21910f980c6fca2b15bee102a7a0d861ed712" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.604ex; height:2.843ex;" alt="{\displaystyle (i,j)}"></span> este <span class="mwe-math-element"><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_{i+j-1},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>+</mo> <mi>j</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S_{i+j-1},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1160d0723f7c8dc8c867f3f97baa432ebf2bdf93" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:6.928ex; height:2.843ex;" alt="{\displaystyle S_{i+j-1},}"></span> este numărul de pavări cu dominouri ale unui diamant aztec de ordinul <span class="mwe-math-element"><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>, care este <span class="mwe-math-element"><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(n+1)/2}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2^{n(n+1)/2}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5e3fac26d8959a35dde9fb2e5bcbfba1bdc3ca55" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.038ex; height:2.843ex;" alt="{\displaystyle 2^{n(n+1)/2}.}"></span><sup id="cite_ref-Eu_Fu_10-0" class="reference"><a href="#cite_note-Eu_Fu-10"><span class="cite-bracket">[</span>10<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 {\begin{vmatrix}S_{1}&S_{2}&\cdots &S_{n}\\S_{2}&S_{3}&\cdots &S_{n+1}\\\vdots &\vdots &\ddots &\vdots \\S_{n}&S_{n+1}&\cdots &S_{2n-1}\end{vmatrix}}=2^{n(n+1)/2}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>|</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> <mtd> <mo>⋯</mo> </mtd> <mtd> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mtd> <mtd> <mo>⋯</mo> </mtd> <mtd> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mo>⋮</mo> </mtd> <mtd> <mo>⋮</mo> </mtd> <mtd> <mo>⋱</mo> </mtd> <mtd> <mo>⋮</mo> </mtd> </mtr> <mtr> <mtd> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mtd> <mtd> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <mo>⋯</mo> </mtd> <mtd> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> </mtd> </mtr> </mtable> <mo>|</mo> </mrow> </mrow> <mo>=</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{vmatrix}S_{1}&S_{2}&\cdots &S_{n}\\S_{2}&S_{3}&\cdots &S_{n+1}\\\vdots &\vdots &\ddots &\vdots \\S_{n}&S_{n+1}&\cdots &S_{2n-1}\end{vmatrix}}=2^{n(n+1)/2}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2942c0112413f4c0ac0d4fd6d27cd78825a7d161" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.505ex; width:37.081ex; height:14.176ex;" alt="{\displaystyle {\begin{vmatrix}S_{1}&S_{2}&\cdots &S_{n}\\S_{2}&S_{3}&\cdots &S_{n+1}\\\vdots &\vdots &\ddots &\vdots \\S_{n}&S_{n+1}&\cdots &S_{2n-1}\end{vmatrix}}=2^{n(n+1)/2}.}"></span></dd></dl> <p>Exemple: </p> <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 {\begin{vmatrix}2\end{vmatrix}}=2=2^{1(2)/2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>|</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>2</mn> </mtd> </mtr> </mtable> <mo>|</mo> </mrow> </mrow> <mo>=</mo> <mn>2</mn> <mo>=</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo stretchy="false">(</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{vmatrix}2\end{vmatrix}}=2=2^{1(2)/2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0e575289988085a6a2d95b82904326f6525f01b8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.529ex; height:3.343ex;" alt="{\displaystyle {\begin{vmatrix}2\end{vmatrix}}=2=2^{1(2)/2}}"></span></li></ul></dd></dl> <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 {\begin{vmatrix}2&6\\6&22\end{vmatrix}}=8=2^{2(3)/2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>|</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>2</mn> </mtd> <mtd> <mn>6</mn> </mtd> </mtr> <mtr> <mtd> <mn>6</mn> </mtd> <mtd> <mn>22</mn> </mtd> </mtr> </mtable> <mo>|</mo> </mrow> </mrow> <mo>=</mo> <mn>8</mn> <mo>=</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mo stretchy="false">(</mo> <mn>3</mn> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{vmatrix}2&6\\6&22\end{vmatrix}}=8=2^{2(3)/2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3715ea4e07db6f3833f8ed6e19dc4be17ce45ec" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:21.176ex; height:6.176ex;" alt="{\displaystyle {\begin{vmatrix}2&6\\6&22\end{vmatrix}}=8=2^{2(3)/2}}"></span></li></ul></dd></dl> <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 {\begin{vmatrix}2&6&22\\6&22&90\\22&90&394\end{vmatrix}}=64=2^{3(4)/2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>|</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>2</mn> </mtd> <mtd> <mn>6</mn> </mtd> <mtd> <mn>22</mn> </mtd> </mtr> <mtr> <mtd> <mn>6</mn> </mtd> <mtd> <mn>22</mn> </mtd> <mtd> <mn>90</mn> </mtd> </mtr> <mtr> <mtd> <mn>22</mn> </mtd> <mtd> <mn>90</mn> </mtd> <mtd> <mn>394</mn> </mtd> </mtr> </mtable> <mo>|</mo> </mrow> </mrow> <mo>=</mo> <mn>64</mn> <mo>=</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> <mo stretchy="false">(</mo> <mn>4</mn> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{vmatrix}2&6&22\\6&22&90\\22&90&394\end{vmatrix}}=64=2^{3(4)/2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/13f0e935c077ed575c6cbff600f618257dd04a24" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:29.311ex; height:9.176ex;" alt="{\displaystyle {\begin{vmatrix}2&6&22\\6&22&90\\22&90&394\end{vmatrix}}=64=2^{3(4)/2}}"></span></li></ul></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Note">Note</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Num%C4%83r_Schr%C3%B6der&veaction=edit&section=7" title="Modifică secțiunea: Note" class="mw-editsection-visualeditor"><span>modificare</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Num%C4%83r_Schr%C3%B6der&action=edit&section=7" title="Edit section's source code: Note"><span>modificare sursă</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="references-small columns references-column-count references-column-count-2" style="-moz-column-count: 2; -webkit-column-count: 2; column-count: 2; list-style-type: decimal;"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-C77-1">^ <a href="#cite_ref-C77_1-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-C77_1-1"><sup><i><b>b</b></i></sup></a> <span class="reference-text">Marius Coman, <i>Enciclopedia matematică a claselor de numere întregi</i>, Columbus, Ohio: Education Publishing, 2013, <a href="/wiki/Special:Referin%C8%9Be_%C3%AEn_c%C4%83r%C8%9Bi/978-1-59973-237-4" title="Special:Referințe în cărți/978-1-59973-237-4">ISBN: 978-1-59973-237-4</a>, p. 77 </span> </li> <li id="cite_note-A006318-2">^ <a href="#cite_ref-A006318_2-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-A006318_2-1"><sup><i><b>b</b></i></sup></a> <span class="reference-text">Șirul <span style="white-space:nowrap;"><a href="//oeis.org/A006318" class="extiw" title="oeis:A006318">A006318</a></span> la <i><a href="/wiki/Enciclopedia_electronic%C4%83_a_%C8%99irurilor_de_numere_%C3%AEntregi" title="Enciclopedia electronică a șirurilor de numere întregi">Enciclopedia electronică a șirurilor de numere întregi</a></i> (OEIS)</span> </li> <li id="cite_note-3"><b><a href="#cite_ref-3">^</a></b> <span class="reference-text"><span style="border:solid 1px #44A; background-color:#EEF; font-family:monospace; color:#008; font-size:0.9em; padding:0px 4px 2px 4px; position:relative; bottom:0.2em; cursor:help;" title="Limba engleză">en</span> <cite class="citation book">Ardila, Federico (<time datetime="2015">2015</time>). „Algebraic and geometric methods in enumerative combinatorics”. <i>Handbook of enumerative combinatorics</i>. Boca Raton, FL: CRC Press. p. 3–172.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Algebraic+and+geometric+methods+in+enumerative+combinatorics&rft.btitle=Handbook+of+enumerative+combinatorics&rft.place=Boca+Raton%2C+FL&rft.pages=3-172&rft.pub=CRC+Press&rft.date=2015&rft.aulast=Ardila&rft.aufirst=Federico&rfr_id=info%3Asid%2Fro.wikipedia.org%3ANum%C4%83r+Schr%C3%B6der" class="Z3988"><span style="display:none;"> </span></span><style data-mw-deduplicate="TemplateStyles:r16236537">.mw-parser-output cite.citation{font-style:inherit}.mw-parser-output .citation q{quotes:"„""”""«""»"}.mw-parser-output .citation .cs1-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Lock-green.svg/9px-Lock-green.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .citation .cs1-lock-limited a,.mw-parser-output .citation .cs1-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/d/d6/Lock-gray-alt-2.svg/9px-Lock-gray-alt-2.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .citation .cs1-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/a/aa/Lock-red-alt-2.svg/9px-Lock-red-alt-2.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration{color:#555}.mw-parser-output .cs1-subscription span,.mw-parser-output .cs1-registration span{border-bottom:1px dotted;cursor:help}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/4/4c/Wikisource-logo.svg/12px-Wikisource-logo.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output code.cs1-code{color:inherit;background:inherit;border:inherit;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;font-size:100%}.mw-parser-output .cs1-visible-error{font-size:100%}.mw-parser-output .cs1-maint{display:none;color:#33aa33;margin-left:0.3em}.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration,.mw-parser-output .cs1-format{font-size:95%}.mw-parser-output .cs1-kern-left,.mw-parser-output .cs1-kern-wl-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right,.mw-parser-output .cs1-kern-wl-right{padding-right:0.2em}</style></span> </li> <li id="cite_note-A001003-4"><b><a href="#cite_ref-A001003_4-0">^</a></b> <span class="reference-text">Șirul <span style="white-space:nowrap;"><a href="//oeis.org/A001003" class="extiw" title="oeis:A001003">A001003</a></span> la <i><a href="/wiki/Enciclopedia_electronic%C4%83_a_%C8%99irurilor_de_numere_%C3%AEntregi" title="Enciclopedia electronică a șirurilor de numere întregi">Enciclopedia electronică a șirurilor de numere întregi</a></i> (OEIS)</span> </li> <li id="cite_note-Bijections_with_Dyck_paths-5">^ <a href="#cite_ref-Bijections_with_Dyck_paths_5-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Bijections_with_Dyck_paths_5-1"><sup><i><b>b</b></i></sup></a> <span class="reference-text"><span style="border:solid 1px #44A; background-color:#EEF; font-family:monospace; color:#008; font-size:0.9em; padding:0px 4px 2px 4px; position:relative; bottom:0.2em; cursor:help;" title="Limba engleză">en</span> <cite class="citation arxiv">Drake, Dan (<time datetime="2010">2010</time>). „Bijections from weighted Dyck paths to Schröder paths”. <a href="/wiki/ArXiv" title="ArXiv">arXiv</a>:<span class="plainlinks"><a rel="nofollow" class="external text" href="//arxiv.org/abs/1006.1959">1006.1959</a> <span typeof="mw:File"><span title="Accesibil gratuit"><img alt="Accesibil gratuit" src="//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Lock-green.svg/9px-Lock-green.svg.png" decoding="async" width="9" height="14" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Lock-green.svg/14px-Lock-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/65/Lock-green.svg/18px-Lock-green.svg.png 2x" data-file-width="512" data-file-height="813" /></span></span></span> [<a rel="nofollow" class="external text" href="//arxiv.org/archive/math.CO">math.CO</a>].</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=preprint&rft.jtitle=arXiv&rft.atitle=Bijections+from+weighted+Dyck+paths+to+Schr%C3%B6der+paths&rft.date=2010&rft_id=info%3Aarxiv%2F1006.1959&rft.aulast=Drake&rft.aufirst=Dan&rfr_id=info%3Asid%2Fro.wikipedia.org%3ANum%C4%83r+Schr%C3%B6der" class="Z3988"><span style="display:none;"> </span></span><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r16236537"></span> </li> <li id="cite_note-Catalan,_Motzkin,_and_Schröder_numbers-6"><b><a href="#cite_ref-Catalan,_Motzkin,_and_Schröder_numbers_6-0">^</a></b> <span class="reference-text"><span style="border:solid 1px #44A; background-color:#EEF; font-family:monospace; color:#008; font-size:0.9em; padding:0px 4px 2px 4px; position:relative; bottom:0.2em; cursor:help;" title="Limba engleză">en</span> <cite class="citation journal">Deng, Eva Y. P.; Yan, Wei-Jun (<time datetime="2008">2008</time>). „Some identities on the Catalan, Motzkin, and Schröder numbers”. <i>Discrete Applied Mathematics</i>. <b>156</b> (166–218X): 2781–2789. <a href="/wiki/Digital_object_identifier" title="Digital object identifier">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fj.dam.2007.11.014">10.1016/j.dam.2007.11.014</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Discrete+Applied+Mathematics&rft.atitle=Some+identities+on+the+Catalan%2C+Motzkin%2C+and+Schr%C3%B6der+numbers&rft.volume=156&rft.issue=166%E2%80%93218X&rft.pages=2781-2789&rft.date=2008&rft_id=info%3Adoi%2F10.1016%2Fj.dam.2007.11.014&rft.aulast=Deng&rft.aufirst=Eva+Y.+P.&rft.au=Yan%2C+Wei-Jun&rfr_id=info%3Asid%2Fro.wikipedia.org%3ANum%C4%83r+Schr%C3%B6der" class="Z3988"><span style="display:none;"> </span></span><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r16236537"></span> </li> <li id="cite_note-A033877-7">^ <a href="#cite_ref-A033877_7-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-A033877_7-1"><sup><i><b>b</b></i></sup></a> <span class="reference-text"><span style="border:solid 1px #44A; background-color:#EEF; font-family:monospace; color:#008; font-size:0.9em; padding:0px 4px 2px 4px; position:relative; bottom:0.2em; cursor:help;" title="Limba engleză">en</span> <cite class="citation web">Sloane, N. J. A. <a rel="nofollow" class="external text" href="https://oeis.org/A033877">„Triangular array associated with Schroeder numbers”</a>. <i>The On-Line Encyclopedia of Integer Sequences</i><span class="reference-accessdate">. Accesat în <time datetime="2018-03-05">5 martie 2018</time></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=The+On-Line+Encyclopedia+of+Integer+Sequences&rft.atitle=Triangular+array+associated+with+Schroeder+numbers&rft.aulast=Sloane&rft.aufirst=N.+J.+A.&rft_id=https%3A%2F%2Foeis.org%2FA033877&rfr_id=info%3Asid%2Fro.wikipedia.org%3ANum%C4%83r+Schr%C3%B6der" class="Z3988"><span style="display:none;"> </span></span><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r16236537"></span> </li> <li id="cite_note-8"><b><a href="#cite_ref-8">^</a></b> <span class="reference-text">Șirul <span style="white-space:nowrap;"><a href="//oeis.org/A033877" class="extiw" title="oeis:A033877">A033877</a></span> la <i><a href="/wiki/Enciclopedia_electronic%C4%83_a_%C8%99irurilor_de_numere_%C3%AEntregi" title="Enciclopedia electronică a șirurilor de numere întregi">Enciclopedia electronică a șirurilor de numere întregi</a></i> (OEIS)</span> </li> <li id="cite_note-Schröder_number_formulas-9">^ <a href="#cite_ref-Schröder_number_formulas_9-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Schröder_number_formulas_9-1"><sup><i><b>b</b></i></sup></a> <span class="reference-text"><span style="border:solid 1px #44A; background-color:#EEF; font-family:monospace; color:#008; font-size:0.9em; padding:0px 4px 2px 4px; position:relative; bottom:0.2em; cursor:help;" title="Limba engleză">en</span> <cite class="citation journal">Oi, Feng; Guo, Bai-Ni (<time datetime="2017">2017</time>). „Some explicit and recursive formulas of the large and little Schröder numbers”. <i>Arab Journal of Mathematical Sciences</i>. <b>23</b> (1319–5166): 141–147. <a href="/wiki/Digital_object_identifier" title="Digital object identifier">doi</a>:<span class="plainlinks"><a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fj.ajmsc.2016.06.002">10.1016/j.ajmsc.2016.06.002</a> <span typeof="mw:File"><span title="Accesibil gratuit"><img alt="Accesibil gratuit" src="//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Lock-green.svg/9px-Lock-green.svg.png" decoding="async" width="9" height="14" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Lock-green.svg/14px-Lock-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/65/Lock-green.svg/18px-Lock-green.svg.png 2x" data-file-width="512" data-file-height="813" /></span></span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Arab+Journal+of+Mathematical+Sciences&rft.atitle=Some+explicit+and+recursive+formulas+of+the+large+and+little+Schr%C3%B6der+numbers&rft.volume=23&rft.issue=1319%E2%80%935166&rft.pages=141-147&rft.date=2017&rft_id=info%3Adoi%2F10.1016%2Fj.ajmsc.2016.06.002&rft.aulast=Oi&rft.aufirst=Feng&rft.au=Guo%2C+Bai-Ni&rfr_id=info%3Asid%2Fro.wikipedia.org%3ANum%C4%83r+Schr%C3%B6der" class="Z3988"><span style="display:none;"> </span></span><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r16236537"></span> </li> <li id="cite_note-Eu_Fu-10"><b><a href="#cite_ref-Eu_Fu_10-0">^</a></b> <span class="reference-text"><span style="border:solid 1px #44A; background-color:#EEF; font-family:monospace; color:#008; font-size:0.9em; padding:0px 4px 2px 4px; position:relative; bottom:0.2em; cursor:help;" title="Limba engleză">en</span> <cite class="citation journal">Eu, Sen-Peng; Fu, Tung-Shan (<time datetime="2005">2005</time>). <a rel="nofollow" class="external text" href="http://www.combinatorics.org/Volume_12/Abstracts/v12i1r18.html">„A simple proof of the Aztec diamond theorem”</a>. <i>Electronic Journal of Combinatorics</i>. <b>12</b> (1077–8926): Research Paper 18, 8.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Electronic+Journal+of+Combinatorics&rft.atitle=A+simple+proof+of+the+Aztec+diamond+theorem&rft.volume=12&rft.issue=1077%E2%80%938926&rft.pages=Research+Paper+18%2C+8&rft.date=2005&rft.aulast=Eu&rft.aufirst=Sen-Peng&rft.au=Fu%2C+Tung-Shan&rft_id=http%3A%2F%2Fwww.combinatorics.org%2FVolume_12%2FAbstracts%2Fv12i1r18.html&rfr_id=info%3Asid%2Fro.wikipedia.org%3ANum%C4%83r+Schr%C3%B6der" class="Z3988"><span style="display:none;"> </span></span><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r16236537"></span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="Lectură_suplimentară"><span id="Lectur.C4.83_suplimentar.C4.83"></span>Lectură suplimentară</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Num%C4%83r_Schr%C3%B6der&veaction=edit&section=8" title="Modifică secțiunea: Lectură suplimentară" class="mw-editsection-visualeditor"><span>modificare</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Num%C4%83r_Schr%C3%B6der&action=edit&section=8" title="Edit section's source code: Lectură suplimentară"><span>modificare sursă</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/w/index.php?title=Richard_P._Stanley&action=edit&redlink=1" class="new" title="Richard P. Stanley — pagină inexistentă">Stanley, Richard P.</a>: <a rel="nofollow" class="external text" href="http://www-math.mit.edu/~rstan/ec/catadd.pdf">Catalan addendum</a> to <i>Enumerative Combinatorics, Volume 2</i></li></ul> <div class="mw-heading mw-heading2"><h2 id="Vezi_și"><span id="Vezi_.C8.99i"></span>Vezi și</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Num%C4%83r_Schr%C3%B6der&veaction=edit&section=9" title="Modifică secțiunea: Vezi și" class="mw-editsection-visualeditor"><span>modificare</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Num%C4%83r_Schr%C3%B6der&action=edit&section=9" title="Edit section's source code: Vezi și"><span>modificare sursă</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Num%C4%83r_Delannoy" title="Număr Delannoy">Număr Delannoy</a></li> <li><a href="/wiki/Num%C4%83r_Motzkin" title="Număr Motzkin">Număr Motzkin</a></li> <li><a href="/wiki/Num%C4%83r_Narayana" title="Număr Narayana">Număr Narayana</a></li> <li><a href="/wiki/Num%C4%83r_Schr%C3%B6der%E2%80%93Hiparh" title="Număr Schröder–Hiparh">Număr Schröder–Hiparh</a></li> <li><a href="/wiki/Num%C4%83r_Catalan" title="Număr Catalan">Număr Catalan</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Legături_externe"><span id="Leg.C4.83turi_externe"></span>Legături externe</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Num%C4%83r_Schr%C3%B6der&veaction=edit&section=10" title="Modifică secțiunea: Legături externe" class="mw-editsection-visualeditor"><span>modificare</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Num%C4%83r_Schr%C3%B6der&action=edit&section=10" title="Edit section's source code: Legături externe"><span>modificare sursă</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><span style="border:solid 1px #44A; background-color:#EEF; font-family:monospace; color:#008; font-size:0.9em; padding:0px 4px 2px 4px; position:relative; bottom:0.2em; cursor:help;" title="Limba engleză">en</span> <cite id="Reference-Mathworld-Schröder_Number"><a href="/wiki/Eric_W._Weisstein" title="Eric W. Weisstein">Eric W. Weisstein</a>, <i><a rel="nofollow" class="external text" href="http://mathworld.wolfram.com/SchroederNumber.html">Schröder Number</a></i> la <a href="/wiki/MathWorld" title="MathWorld">MathWorld</a>.</cite></li></ul>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Adus de la <a dir="ltr" href="https://ro.wikipedia.org/w/index.php?title=Număr_Schröder&oldid=15806940">https://ro.wikipedia.org/w/index.php?title=Număr_Schröder&oldid=15806940</a></div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Special:Categorii" title="Special:Categorii">Categorii</a>: <ul><li><a href="/wiki/Categorie:%C8%98iruri_de_numere_%C3%AEntregi" title="Categorie:Șiruri de numere întregi">Șiruri de numere întregi</a></li><li><a href="/wiki/Categorie:Combinatoric%C4%83" title="Categorie:Combinatorică">Combinatorică</a></li></ul></div><div id="mw-hidden-catlinks" class="mw-hidden-catlinks mw-hidden-cats-hidden">Categorie ascunsă: <ul><li><a href="/wiki/Categorie:Pagini_cu_note_pe_2_coloane" title="Categorie:Pagini cu note pe 2 coloane">Pagini cu note pe 2 coloane</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"> Ultima editare a paginii a fost efectuată la 5 august 2023, ora 17:22.</li> <li id="footer-info-copyright">Acest text este disponibil sub licența <a rel="nofollow" class="external text" href="https://creativecommons.org/licenses/by-sa/4.0/deed.ro">Creative Commons cu atribuire și distribuire în condiții identice</a>; pot exista și clauze suplimentare. 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