ff1588d761d55f48f61a52249e1dcc24

Micron Document

<!DOCTYPE html>

<title>Bell-Polynom</title>

</head>

<h1 id="firstHeading" class="firstHeading mw-first-heading">Bell-Polynom</h1>

</header>

</div>

<div id="mw-content-text" class="mw-body-content mw-content-ltr" lang="de" dir="ltr"><div class="mw-content-ltr mw-parser-output" lang="de" dir="ltr">Im mathematischen Teilgebiet der <a href="Kombinatorik" title="Kombinatorik">Kombinatorik</a> bezeichnen die Bell-Polynome, benannt nach <a href="Eric_Temple_Bell" class="mw-redirect" title="Eric Temple Bell">Eric Temple Bell</a>, folgende dreieckige Anordnung von <a href="Polynom" title="Polynom">Polynomen</a>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<mfrac>

<mrow>

</mrow>

<mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mfrac>

</mrow>

<msup>

<mrow>

<mfrac>

<msub>

</mrow>

</msub>

<mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</msup>

<msup>

<mrow>

<mfrac>

<msub>

</mrow>

</msub>

<mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</msup>

<msup>

<mrow>

<mfrac>

<msub>

</mrow>

</msub>

<mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle B_{n,k}(x_{1},x_{2},\dots ,x_{n-k+1})=\sum {\frac {n!}{j_{1}!j_{2}!\cdots j_{n-k+1}!}}\left({\frac {x_{1}}{1!}}\right)^{j_{1}}\left({\frac {x_{2}}{2!}}\right)^{j_{2}}\cdots \left({\frac {x_{n-k+1}}{(n-k+1)!}}\right)^{j_{n-k+1}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e4d919101f89b6e893c9acedc9bf5c9bc9c16863.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:87.868ex; height:6.676ex;" alt="{\displaystyle B_{n,k}(x_{1},x_{2},\dots ,x_{n-k+1})=\sum {\frac {n!}{j_{1}!j_{2}!\cdots j_{n-k+1}!}}\left({\frac {x_{1}}{1!}}\right)^{j_{1}}\left({\frac {x_{2}}{2!}}\right)^{j_{2}}\cdots \left({\frac {x_{n-k+1}}{(n-k+1)!}}\right)^{j_{n-k+1}}}" loading="lazy"> ,</dd></dl>

wobei die Summe über alle Sequenzen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle j_{1},j_{2},\dots ,j_{n-k+1}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle j_{1},j_{2},\dots ,j_{n-k+1}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2bd06633eed0231d211fedcf0bc0fef05f7f5d1d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.027ex; width:16.676ex; height:2.509ex;" alt="{\displaystyle j_{1},j_{2},\dots ,j_{n-k+1}}" loading="lazy"> von nicht-negativen ganzen Zahlen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle j_{i}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle j_{i}}</annotation>

</semantics>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle j_{1}+j_{2}+j_{3}+\cdots =k}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e9004c335fb943da14d547aa56393463d9881af0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.027ex; width:21.618ex; height:2.509ex;" alt="{\displaystyle j_{1}+j_{2}+j_{3}+\cdots =k}" loading="lazy">       und       <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1\,j_{1}\;+\;2\,j_{2}\;+\;3\,j_{3}\;+\cdots =n}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 1\,j_{1}\;+\;2\,j_{2}\;+\;3\,j_{3}\;+\cdots =n}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/48ef4fd23dc69e6d9985931f75a5a841d0c00112.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:29.649ex; height:2.509ex;" alt="{\displaystyle 1\,j_{1}\;+\;2\,j_{2}\;+\;3\,j_{3}\;+\cdots =n}" loading="lazy"> .</dd></dl>

Das Bell-Polynom <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B_{n,k}(x_{1},x_{2},\dots ,x_{n-k+1})}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle B_{n,k}(x_{1},x_{2},\dots ,x_{n-k+1})}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4dbd1af656b8288b6d61e72f68be99c18cc96282.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:23.869ex; height:3.009ex;" alt="{\displaystyle B_{n,k}(x_{1},x_{2},\dots ,x_{n-k+1})}" loading="lazy"> ist ein <a href="Polynom#Polynome_in_mehreren_Unbestimmten" title="Polynom">Polynom</a> in den Variablen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{1},x_{2},\dots ,x_{n-k+1}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x_{1},x_{2},\dots ,x_{n-k+1}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/cb56314566a8607c119b087c8399fde48f925ac0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:17.764ex; height:2.009ex;" alt="{\displaystyle x_{1},x_{2},\dots ,x_{n-k+1}}" loading="lazy">. Seine Koeffizienten sind <a href="Ganze_Zahl" title="Ganze Zahl">ganze Zahlen</a>.

<div class="mw-heading mw-heading2"><h2 id="Vollständige_Bell-Polynome">Vollständige Bell-Polynome</h2></div>

Die Summe

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</munderover>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle B_{n}(x_{1},\dots ,x_{n})=\sum _{k=1}^{n}B_{n,k}(x_{1},x_{2},\dots ,x_{n-k+1})}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/76991d0ee74bbcc3c1dd06b481c3136c873d28a4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:45.612ex; height:6.843ex;" alt="{\displaystyle B_{n}(x_{1},\dots ,x_{n})=\sum _{k=1}^{n}B_{n,k}(x_{1},x_{2},\dots ,x_{n-k+1})}" loading="lazy"></dd></dl>

wird manchmal als <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" 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}" loading="lazy">tes vollständiges Bell-Polynom bezeichnet. Zur besseren Abgrenzung gegenüber den vollständigen Bell-Polynomen, werden die oben definierten Polynome <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B_{n,k}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle B_{n,k}}</annotation>

</semantics>

Die vollständigen Bell-Polynome genügen folgender Gleichung

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<mrow>

<mtr>

<mtd>

<msub>

</mrow>

</msub>

</mtd>

<mtd>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

<msub>

</mrow>

</msub>

</mtd>

<mtd>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

<msub>

</mrow>

</msub>

</mtd>

<mtd>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

<msub>

</mrow>

</msub>

</mtd>

<mtd>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

<msub>

</mrow>

</msub>

</mtd>

<mtd>

</mtd>

<mtd>

<msub>

</mrow>

</msub>

</mtd>

</mtr>

<mtr>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

<msub>

</mrow>

</msub>

</mtd>

<mtd>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

<msub>

</mrow>

</msub>

</mtd>

<mtd>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

<msub>

</mrow>

</msub>

</mtd>

<mtd>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

<msub>

</mrow>

</msub>

</mtd>

<mtd>

</mtd>

<mtd>

<msub>

</mrow>

</msub>

</mtd>

</mtr>

<mtr>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

<msub>

</mrow>

</msub>

</mtd>

<mtd>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

<msub>

</mrow>

</msub>

</mtd>

<mtd>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

<msub>

</mrow>

</msub>

</mtd>

<mtd>

</mtd>

<mtd>

<msub>

</mrow>

</msub>

</mtd>

</mtr>

<mtr>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

<msub>

</mrow>

</msub>

</mtd>

<mtd>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

<msub>

</mrow>

</msub>

</mtd>

<mtd>

</mtd>

<mtd>

<msub>

</mrow>

</msub>

</mtd>

</mtr>

<mtr>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

<msub>

</mrow>

</msub>

</mtd>

<mtd>

</mtd>

<mtd>

<msub>

</mrow>

</msub>

</mtd>

</mtr>

<mtr>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

<mtr>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

<msub>

</mrow>

</msub>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle B_{n}(x_{1},\dots ,x_{n})=\det {\begin{bmatrix}x_{1}&{\binom {n-1}{1}}x_{2}&{\binom {n-1}{2}}x_{3}&{\binom {n-1}{3}}x_{4}&{\binom {n-1}{4}}x_{5}&\cdots &\;\;x_{n}\\\\-1&x_{1}&{\binom {n-2}{1}}x_{2}&{\binom {n-2}{2}}x_{3}&{\binom {n-2}{3}}x_{4}&\cdots &\;\;x_{n-1}\\\\0&-1&x_{1}&{\binom {n-3}{1}}x_{2}&{\binom {n-3}{2}}x_{3}&\cdots &\;\;x_{n-2}\\\\0&0&-1&x_{1}&{\binom {n-4}{1}}x_{2}&\cdots &\;\;x_{n-3}\\\\0&0&0&-1&x_{1}&\cdots &\;\;x_{n-4}\\\\\vdots &\vdots &\vdots &\vdots &\ddots &\ddots &\;\;\vdots \\\\0&0&0&0&\cdots &-1&\;\;x_{1}\end{bmatrix}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/976f14f6db729ce3967fd4b6dcf1200dc17df82d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -22.505ex; width:81.695ex; height:46.176ex;" alt="{\displaystyle B_{n}(x_{1},\dots ,x_{n})=\det {\begin{bmatrix}x_{1}&{\binom {n-1}{1}}x_{2}&{\binom {n-1}{2}}x_{3}&{\binom {n-1}{3}}x_{4}&{\binom {n-1}{4}}x_{5}&\cdots &\;\;x_{n}\\\\-1&x_{1}&{\binom {n-2}{1}}x_{2}&{\binom {n-2}{2}}x_{3}&{\binom {n-2}{3}}x_{4}&\cdots &\;\;x_{n-1}\\\\0&-1&x_{1}&{\binom {n-3}{1}}x_{2}&{\binom {n-3}{2}}x_{3}&\cdots &\;\;x_{n-2}\\\\0&0&-1&x_{1}&{\binom {n-4}{1}}x_{2}&\cdots &\;\;x_{n-3}\\\\0&0&0&-1&x_{1}&\cdots &\;\;x_{n-4}\\\\\vdots &\vdots &\vdots &\vdots &\ddots &\ddots &\;\;\vdots \\\\0&0&0&0&\cdots &-1&\;\;x_{1}\end{bmatrix}}}" loading="lazy"></dd></dl>

<div class="mw-heading mw-heading3"><h3 id="Beispiele">Beispiele</h3></div>

Die ersten vollständigen Bell-Polynome lauten:

<mtr>

<mtd>

<msub>

</mrow>

</msub>

</mrow>

</mtd>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mtd>

<mtd>

<msub>

</mrow>

</msub>

</mtd>

</mtr>

<mtr>

<mtd>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mtd>

<mtd>

</mrow>

</mrow>

</msubsup>

<msub>

</mrow>

</msub>

</mtd>

</mtr>

<mtr>

<mtd>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mtd>

<mtd>

</mrow>

</mrow>

</msubsup>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mtd>

</mtr>

<mtr>

<mtd>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mtd>

<mtd>

</mrow>

</mrow>

</msubsup>

</mrow>

</mrow>

</msubsup>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</msubsup>

<msub>

</mrow>

</msub>

</mtd>

</mtr>

<mtr>

<mtd>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mtd>

<mtd>

</mrow>

</mrow>

</msubsup>

</mrow>

</mrow>

</msubsup>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</msubsup>

</mrow>

</mrow>

</msubsup>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mtd>

</mtr>

<mtr>

<mtd>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mtd>

<mtd>

</mrow>

</mrow>

</msubsup>

</mrow>

</mrow>

</msubsup>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</msubsup>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</msubsup>

</mrow>

</mrow>

</msubsup>

</mrow>

</mrow>

</msubsup>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mtd>

</mtr>

<mtr>

<mtd>

</mrow>

</mrow>

</mrow>

</msubsup>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</msubsup>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mtd>

</mtr>

<mtr>

<mtd>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mtd>

<mtd>

</mrow>

</mrow>

</msubsup>

</mrow>

</mrow>

</msubsup>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</msubsup>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</msubsup>

</mrow>

</mrow>

</msubsup>

</mrow>

</mrow>

</msubsup>

<msub>

</mrow>

</msub>

</mtd>

</mtr>

<mtr>

<mtd>

</mrow>

</mrow>

</mrow>

</msubsup>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</msubsup>

</mrow>

</mrow>

</msubsup>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mtd>

</mtr>

<mtr>

<mtd>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</msubsup>

</mrow>

</mrow>

</msubsup>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}B_{0}={}&1,\\[8pt]B_{1}(x_{1})={}&x_{1},\\[8pt]B_{2}(x_{1},x_{2})={}&x_{1}^{2}+x_{2},\\[8pt]B_{3}(x_{1},x_{2},x_{3})={}&x_{1}^{3}+3x_{1}x_{2}+x_{3},\\[8pt]B_{4}(x_{1},x_{2},x_{3},x_{4})={}&x_{1}^{4}+6x_{1}^{2}x_{2}+4x_{1}x_{3}+3x_{2}^{2}+x_{4},\\[8pt]B_{5}(x_{1},x_{2},x_{3},x_{4},x_{5})={}&x_{1}^{5}+10x_{1}^{3}x_{2}+15x_{1}x_{2}^{2}+10x_{1}^{2}x_{3}+10x_{2}x_{3}+5x_{1}x_{4}+x_{5}\\[8pt]B_{6}(x_{1},x_{2},x_{3},x_{4},x_{5},x_{6})={}&x_{1}^{6}+15x_{1}^{4}x_{2}+20x_{1}^{3}x_{3}+45x_{1}^{2}x_{2}^{2}+15x_{2}^{3}+60x_{1}x_{2}x_{3}\\&{}+15x_{1}^{2}x_{4}+10x_{3}^{2}+15x_{2}x_{4}+6x_{1}x_{5}+x_{6},\\[8pt]B_{7}(x_{1},x_{2},x_{3},x_{4},x_{5},x_{6},x_{7})={}&x_{1}^{7}+21x_{1}^{5}x_{2}+35x_{1}^{4}x_{3}+105x_{1}^{3}x_{2}^{2}+35x_{1}^{3}x_{4}\\&{}+210x_{1}^{2}x_{2}x_{3}+105x_{1}x_{2}^{3}+21x_{1}^{2}x_{5}+105x_{1}x_{2}x_{4}\\&{}+70x_{1}x_{3}^{2}+105x_{2}^{2}x_{3}+7x_{1}x_{6}+21x_{2}x_{5}+35x_{3}x_{4}+x_{7}.\end{aligned}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/73120ef95cd07c039c872a0e919f7e6484eb936c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -23.684ex; margin-bottom: -0.321ex; width:87.48ex; height:49.176ex;" alt="{\displaystyle {\begin{aligned}B_{0}={}&1,\\[8pt]B_{1}(x_{1})={}&x_{1},\\[8pt]B_{2}(x_{1},x_{2})={}&x_{1}^{2}+x_{2},\\[8pt]B_{3}(x_{1},x_{2},x_{3})={}&x_{1}^{3}+3x_{1}x_{2}+x_{3},\\[8pt]B_{4}(x_{1},x_{2},x_{3},x_{4})={}&x_{1}^{4}+6x_{1}^{2}x_{2}+4x_{1}x_{3}+3x_{2}^{2}+x_{4},\\[8pt]B_{5}(x_{1},x_{2},x_{3},x_{4},x_{5})={}&x_{1}^{5}+10x_{1}^{3}x_{2}+15x_{1}x_{2}^{2}+10x_{1}^{2}x_{3}+10x_{2}x_{3}+5x_{1}x_{4}+x_{5}\\[8pt]B_{6}(x_{1},x_{2},x_{3},x_{4},x_{5},x_{6})={}&x_{1}^{6}+15x_{1}^{4}x_{2}+20x_{1}^{3}x_{3}+45x_{1}^{2}x_{2}^{2}+15x_{2}^{3}+60x_{1}x_{2}x_{3}\\&{}+15x_{1}^{2}x_{4}+10x_{3}^{2}+15x_{2}x_{4}+6x_{1}x_{5}+x_{6},\\[8pt]B_{7}(x_{1},x_{2},x_{3},x_{4},x_{5},x_{6},x_{7})={}&x_{1}^{7}+21x_{1}^{5}x_{2}+35x_{1}^{4}x_{3}+105x_{1}^{3}x_{2}^{2}+35x_{1}^{3}x_{4}\\&{}+210x_{1}^{2}x_{2}x_{3}+105x_{1}x_{2}^{3}+21x_{1}^{2}x_{5}+105x_{1}x_{2}x_{4}\\&{}+70x_{1}x_{3}^{2}+105x_{2}^{2}x_{3}+7x_{1}x_{6}+21x_{2}x_{5}+35x_{3}x_{4}+x_{7}.\end{aligned}}}" loading="lazy"></dd></dl>

<div class="mw-heading mw-heading2"><h2 id="Rekursive_Darstellungen">Rekursive Darstellungen</h2></div>

Eine <a href="Rekursion" title="Rekursion">rekursive Definition</a> der Bell-Polynome ist:

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle B_{0,0}()}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0ac76d0e2df4e4de61ada1bcff8959fc5af146a6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:5.907ex; height:3.009ex;" alt="{\displaystyle B_{0,0}()}" loading="lazy"></td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle =1}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/282a76fe69ce05e31352dfd19b7700eb784fb3f8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.616ex; height:2.176ex;" alt="{\displaystyle =1}" loading="lazy"> ,

</td></tr>

<tr>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle B_{n,0}(x_{1},\dots ,x_{n+1})}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f009a4c4c1f949d72471b362261f3a3e1a627853.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:18.282ex; height:3.009ex;" alt="{\displaystyle B_{n,0}(x_{1},\dots ,x_{n+1})}" loading="lazy"></td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle =0}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6dc9e66de468806365c20e32e83456cc526ce29e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.616ex; height:2.176ex;" alt="{\displaystyle =0}" loading="lazy"></td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n\geq 1}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d8ce9ce38d06f6bf5a3fe063118c09c2b6202bfe.svg" 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 1}" loading="lazy"> ,

</td></tr>

<tr>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle B_{n,k}(x_{1},\dots ,x_{n-k+1})}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7b8d94b3715a13f3044daaf19b149e8bd4b77e3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:20.451ex; height:3.009ex;" alt="{\displaystyle B_{n,k}(x_{1},\dots ,x_{n-k+1})}" loading="lazy"></td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle =0}</annotation>

</semantics>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n\geq 0,k>n}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/65affb6721d057f6e68b1e7a16cec5c90ca31e6b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.394ex; height:2.509ex;" alt="{\displaystyle n\geq 0,k>n}" loading="lazy">

</td></tr></tbody></table></dd></dl>

und

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle B_{n,k}(x_{1},\dots ,x_{n-k+1})}</annotation>

</semantics>

</mrow>

</mrow>

</munderover>

<mrow>

</mrow>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle =\sum _{i=1}^{n-k+1}{\binom {n-1}{i-1}}B_{n-i,k-1}(x_{1},\dots ,x_{n-i-k+2})\,x_{i}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/dd34bb610e66e63de451d34b977f4043825daa05.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:45.641ex; height:7.343ex;" alt="{\displaystyle =\sum _{i=1}^{n-k+1}{\binom {n-1}{i-1}}B_{n-i,k-1}(x_{1},\dots ,x_{n-i-k+2})\,x_{i}}" loading="lazy"></td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n\geq 1,k\leq n}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/814761ca5704d6d8357b53ccc94714715cee7ee0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.394ex; height:2.509ex;" alt="{\displaystyle n\geq 1,k\leq n}" loading="lazy"> .

</td></tr></tbody></table></dd></dl>

Die vollständigen Bell-Polynome können folgendermaßen rekursiv definiert werden

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle B_{0}(x_{1},\ldots ,x_{n+1})=1}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/54c288036822258ae71c8828d2fe7bec54737b2c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.099ex; height:2.843ex;" alt="{\displaystyle B_{0}(x_{1},\ldots ,x_{n+1})=1}" loading="lazy">

</td></tr></tbody></table></dd></dl>

und

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</munderover>

<mrow>

</mrow>

</mfrac>

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle B_{n+1}(x_{1},\ldots ,x_{n+1})=\sum _{i=0}^{n}{\binom {n}{i}}B_{n-i}(x_{1},\ldots ,x_{n-i})\,x_{i+1}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/75873bab6dec5b4b8542b47d25511c3a5c67c693.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:53.971ex; height:6.843ex;" alt="{\displaystyle B_{n+1}(x_{1},\ldots ,x_{n+1})=\sum _{i=0}^{n}{\binom {n}{i}}B_{n-i}(x_{1},\ldots ,x_{n-i})\,x_{i+1}}" loading="lazy"></td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n\geq 0}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ce8a1b7b3bc3c790054d93629fc3b08cd1da1fd0.svg" 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 0}" loading="lazy"> .

</td></tr></tbody></table></dd></dl>

Sie erfüllen auch die folgende rekursive Differentialformel:

<mtr>

<mtd>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

<mrow>

</mrow>

</mrow>

</munderover>

</mrow>

</mtd>

<mtd>

</mrow>

</mrow>

</munderover>

<mrow>

</mrow>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<mfrac>

<mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

</mfrac>

</mrow>

</mtd>

</mtr>

<mtr>

<mtd>

<mrow>

<mrow>

</mrow>

</mrow>

</mrow>

</munderover>

</mrow>

</mrow>

</munderover>

<mfrac>

<msub>

</mrow>

</msub>

<mrow>

</mrow>

</mfrac>

</mrow>

</mfrac>

</mrow>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

<mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mfrac>

</mrow>

</mrow>

</mtd>

</mtr>

<mtr>

<mtd>

<mrow>

<mrow>

</mrow>

</mrow>

</mrow>

</munderover>

<msub>

</mrow>

</msub>

<mfrac>

<mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

</mfrac>

</mrow>

</mrow>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}B_{n}(x_{1},\ldots ,x_{n})={\frac {1}{n-1}}\left[\sum _{i=2}^{n}\right.&\sum _{j=1}^{i-1}(i-1){\binom {i-2}{j-1}}x_{j}x_{i-j}{\frac {\partial B_{n-1}(x_{1},\dots ,x_{n-1})}{\partial x_{i-1}}}\\[5pt]&\left.{}+\sum _{i=2}^{n}\sum _{j=1}^{i-1}{\frac {x_{i+1}}{\binom {i}{j}}}{\frac {\partial ^{2}B_{n-1}(x_{1},\dots ,x_{n-1})}{\partial x_{j}\partial x_{i-j}}}\right.\\[5pt]&\left.{}+\sum _{i=2}^{n}x_{i}{\frac {\partial B_{n-1}(x_{1},\dots ,x_{n-1})}{\partial x_{i-1}}}\right].\end{aligned}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7f8063f0be30dd504b0ae58533a3ce24749fcc34.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -12.171ex; width:76.404ex; height:25.509ex;" alt="{\displaystyle {\begin{aligned}B_{n}(x_{1},\ldots ,x_{n})={\frac {1}{n-1}}\left[\sum _{i=2}^{n}\right.&\sum _{j=1}^{i-1}(i-1){\binom {i-2}{j-1}}x_{j}x_{i-j}{\frac {\partial B_{n-1}(x_{1},\dots ,x_{n-1})}{\partial x_{i-1}}}\\[5pt]&\left.{}+\sum _{i=2}^{n}\sum _{j=1}^{i-1}{\frac {x_{i+1}}{\binom {i}{j}}}{\frac {\partial ^{2}B_{n-1}(x_{1},\dots ,x_{n-1})}{\partial x_{j}\partial x_{i-j}}}\right.\\[5pt]&\left.{}+\sum _{i=2}^{n}x_{i}{\frac {\partial B_{n-1}(x_{1},\dots ,x_{n-1})}{\partial x_{i-1}}}\right].\end{aligned}}}" loading="lazy"></dd></dl>

<div class="mw-heading mw-heading2"><h2 id="Kombinatorische_Bedeutung">Kombinatorische Bedeutung</h2></div>

Gegeben sei eine nicht-negative ganze Zahl <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\in \mathbb {N} _{0}}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n\in \mathbb {N} _{0}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0cf5c8ad993619a325ca57a25c22cdc75a460f88.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.968ex; height:2.509ex;" alt="{\displaystyle n\in \mathbb {N} _{0}}" loading="lazy"> als Elementeanzahl der zu partitionierenden Menge.

Wird die ganze Zahl (= eine Menge der Größe) <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" 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}" loading="lazy"> in eine Summe von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle k}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c3c9a2c7b599b37105512c5d570edc034056dd40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" loading="lazy"> Summanden (= Partitionen) <a href="Partitionsfunktion" title="Partitionsfunktion">zerlegt</a>, in der der Summand (= die Partitionsgröße) 1 <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle j_{1}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle j_{1}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0a671019a672cd149a8d1584861e2f8fe9728a9e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.027ex; width:2.039ex; height:2.509ex;" alt="{\displaystyle j_{1}}" loading="lazy"> mal, die 2 <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle j_{2}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle j_{2}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9f29268662641c1e61d6faba764a493c74ba962b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.027ex; width:2.039ex; height:2.509ex;" alt="{\displaystyle j_{2}}" loading="lazy"> mal und der Summand <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle i}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/add78d8608ad86e54951b8c8bd6c8d8416533d20.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}" loading="lazy"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle j_{i}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle j_{i}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f2c4b258245d4ce139e3a77a525eb7efd4a6e270.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.027ex; width:1.784ex; height:2.509ex;" alt="{\displaystyle j_{i}}" loading="lazy"> mal vorkommt, dann entspricht die Anzahl der möglichen <a href="Partition_(Mengenlehre)" title="Partition (Mengenlehre)">Partitionierungen</a>, die mit einer <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" 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}" loading="lazy">-elementigen Menge gebildet werden können, dem den <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle k}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c3c9a2c7b599b37105512c5d570edc034056dd40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" loading="lazy"> Partitionsgrößen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (1,2,\dots ,k)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (1,2,\dots ,k)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1482115d6f453a4ff9a1e2cc9b36c499dbeb5333.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.558ex; height:2.843ex;" alt="{\displaystyle (1,2,\dots ,k)}" loading="lazy"> zuzuordnenden Koeffizienten des Monoms <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{1}^{j_{1}}\cdots x_{k}^{j_{k}}}">

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</msubsup>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x_{1}^{j_{1}}\cdots x_{k}^{j_{k}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/74dbf04d4223fdf1e2b714451757d19f7c1839ae.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:9.667ex; height:3.509ex;" alt="{\displaystyle x_{1}^{j_{1}}\cdots x_{k}^{j_{k}}}" loading="lazy"> im Bell-Polynom. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B_{n,k}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle B_{n,k}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b70a55a02dca54fe8db8174fdf040477feff8d0f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:4.296ex; height:2.843ex;" alt="{\displaystyle B_{n,k}}" loading="lazy"> ist dann das Polynom aus allen Monomen mit dem <a href="Polynom#Polynome_in_mehreren_Unbestimmten" title="Polynom">Totalgrad</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k=j_{1}+j_{2}+j_{3}+\cdots =k}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle k=j_{1}+j_{2}+j_{3}+\cdots =k}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/49be7ce1db3834e14b4ce9a1fc581ed104c80a43.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:25.901ex; height:2.509ex;" alt="{\displaystyle k=j_{1}+j_{2}+j_{3}+\cdots =k}" loading="lazy"> und der Mengengröße <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n=1\,j_{1}\;+\;2\,j_{2}\;+\;3\,j_{3}\;+\cdots }">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n=1\,j_{1}\;+\;2\,j_{2}\;+\;3\,j_{3}\;+\cdots }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f9dcecbd0dfd3158cab58eab222727430c6ae5a2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:29.649ex; height:2.509ex;" alt="{\displaystyle n=1\,j_{1}\;+\;2\,j_{2}\;+\;3\,j_{3}\;+\cdots }" loading="lazy">.

<table>

<td colspan="2">Die Namen (eigentlich: die Nummern) der Unbestimmten</td>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x_{1},}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fb7118b371b245f6611bd7a6c8ee319dbdc32825.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.031ex; height:2.009ex;" alt="{\displaystyle x_{1},}" loading="lazy"></td>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x_{2},}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e66e41f50371ed4b1910b485f2e2d0eb28f11533.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.031ex; height:2.009ex;" alt="{\displaystyle x_{2},}" loading="lazy"></td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \dots ,}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f4dec7c45e485b15b103a2b0974678a3d49b9032.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.757ex; height:1.176ex;" alt="{\displaystyle \dots ,}" loading="lazy"></td>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x_{n-k+1}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fc8563ab7c1ba08caaf3befdbb2d6af3b667227a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.784ex; height:2.009ex;" alt="{\displaystyle x_{n-k+1}}" loading="lazy">

</td></tr>

<tr>

<td colspan="2">fungieren dabei nur als Pfosten zum Anheften der Anzahl</td>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle j_{1},}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/443605ef08d68ca0c6d00b1fafb715bd111037a1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.027ex; width:2.686ex; height:2.509ex;" alt="{\displaystyle j_{1},}" loading="lazy"></td>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle j_{2},}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/00d8047b1d9a39aa0faa315f224a6958ac045041.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.027ex; width:2.686ex; height:2.509ex;" alt="{\displaystyle j_{2},}" loading="lazy"></td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \dots ,}</annotation>

</semantics>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle j_{n-k+1}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d6440ee6ce1e368d03042c7105bd097b6444458d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.027ex; width:6.439ex; height:2.509ex;" alt="{\displaystyle j_{n-k+1}}" loading="lazy">

</td></tr>

<tr>

<td>der Partitionen in der Partitionierung, die genau</td>

<td style="width:11em;text-align:right">Summand <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \in \{}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \in \{}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7696ceb158642d914c871cb86b1eb18b60487626.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.358ex; height:2.843ex;" alt="{\displaystyle \in \{}" loading="lazy"></td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 1,}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9cc5fd8163a83100c5330622e9e317fa4e872403.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.809ex; height:2.509ex;" alt="{\displaystyle 1,}" loading="lazy"></td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 2,}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/993d53082bc05c266933af9a892e1ce2128547cd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.809ex; height:2.509ex;" alt="{\displaystyle 2,}" loading="lazy"></td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \dots ,}</annotation>

</semantics>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n\!-\!k\!+\!1}</annotation>

</semantics>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2cf208e5d370391e66767f13641bd5ee6ad93825.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.162ex; height:2.843ex;" alt="{\displaystyle \}}" loading="lazy"></td>

<td>  Elemente haben sollen,

</td></tr>

<tr>

<td colspan="6">als Exponent der Potenz <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{i}^{j_{i}}}">

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x_{i}^{j_{i}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5a1c4b92f24a93ad87aafc30467d77bdb60960f5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.864ex; height:3.509ex;" alt="{\displaystyle x_{i}^{j_{i}}}" loading="lazy">.

</td></tr></tbody></table>

Ein Exponent 1 wird normalerweise nicht notiert.

Ist der Exponent 0, dann wird die ganze Potenz <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{i}^{0}}">

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x_{i}^{0}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6319e9660975eba91bf542b34f0d68748f6c450e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.384ex; height:3.176ex;" alt="{\displaystyle x_{i}^{0}}" loading="lazy"> unterdrückt.

Die größte Partitionsgröße bei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle k}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c3c9a2c7b599b37105512c5d570edc034056dd40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" loading="lazy"> Partitionen ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\!-\!k\!+\!1}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n\!-\!k\!+\!1}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f500ce8de152d4233444269ce20076147ebb6d67.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:7.901ex; height:2.343ex;" alt="{\displaystyle n\!-\!k\!+\!1}" loading="lazy">, welche Partitionsgröße dann genau <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle j_{n-k+1}=1}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle j_{n-k+1}=1}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/23112e06e093f138692dd43578d9d711ee8eac16.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.027ex; width:10.7ex; height:2.509ex;" alt="{\displaystyle j_{n-k+1}=1}" loading="lazy"> mal vorkommt. Die kleinste Partitionsgröße (= 1) kommt dann in dieser Partitionierung genau <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle j_{1}=k-1}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle j_{1}=k-1}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5e0ecb6c0abe68a1a57416a689c6ce21aab33d14.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.027ex; width:10.352ex; height:2.509ex;" alt="{\displaystyle j_{1}=k-1}" loading="lazy"> mal vor.

Die bevorzugte Reihenfolge der Monome im Bell-Polynom ist die lexikographisch aufsteigende mit niedrigstem Rang für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{1}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x_{1}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a8788bf85d532fa88d1fb25eff6ae382a601c308.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.384ex; height:2.009ex;" alt="{\displaystyle x_{1}}" loading="lazy">, also <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{1}^{\,2}x_{2}=x_{1}x_{1}x_{2}}">

</mrow>

</mrow>

</msubsup>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x_{1}^{\,2}x_{2}=x_{1}x_{1}x_{2}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e1380501c4936acd111f353630b11b2d11bddb45.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:15.405ex; height:3.176ex;" alt="{\displaystyle x_{1}^{\,2}x_{2}=x_{1}x_{1}x_{2}}" loading="lazy"> kommt vor <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{1}x_{2}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x_{1}x_{2}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1b72aaacad232ae448753193b1f80b434bc61b88.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.768ex; height:2.009ex;" alt="{\displaystyle x_{1}x_{2}}" loading="lazy"> kommt vor <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{2}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x_{2}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d7af1b928f06e4c7e3e8ebfd60704656719bd766.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.384ex; height:2.009ex;" alt="{\displaystyle x_{2}}" loading="lazy">.

<dl><dt>Beispiele</dt></dl>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle B_{n,1}(x_{1},\dots ,x_{n})=x_{n}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/189f3cacefb02f3bd725a561dcd8d486fb57a884.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:21.828ex; height:3.009ex;" alt="{\displaystyle B_{n,1}(x_{1},\dots ,x_{n})=x_{n}}" loading="lazy"> für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\geq 1}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n\geq 1}</annotation>

</semantics>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle B_{n,k}(x_{1},\dots ,x_{n-k+1})=0}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e69a47f0ad11d689238c337da67e9aee91bffaac.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:24.712ex; height:3.009ex;" alt="{\displaystyle B_{n,k}(x_{1},\dots ,x_{n-k+1})=0}" loading="lazy"> für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k>n}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle k>n}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/66e81682bf174c978e9008ffb557ba4da2cf7478.svg" 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}" loading="lazy"> .</li>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle B_{n,n}(x_{1},\dots ,x_{n-k+1})=x_{1}^{n}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a148b80a1a1b24286e29b3d595bc9b1889eb9297.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:26.228ex; height:3.009ex;" alt="{\displaystyle B_{n,n}(x_{1},\dots ,x_{n-k+1})=x_{1}^{n}}" loading="lazy"> für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\geq 1,k\leq n}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n\geq 1,k\leq n}</annotation>

</semantics>

<li>Ferner ist</li></ul>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle B_{6,2}(x_{1},x_{2},x_{3},x_{4},x_{5})=6x_{1}x_{5}+15x_{2}x_{4}+10x_{3}^{2}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/feb1918630df697231e26d61f02fd1d49c4db972.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:48.473ex; height:3.176ex;" alt="{\displaystyle B_{6,2}(x_{1},x_{2},x_{3},x_{4},x_{5})=6x_{1}x_{5}+15x_{2}x_{4}+10x_{3}^{2}}" loading="lazy"> ,</dd></dl></dd>

<dd>da es

<dl><dd>(Monom <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 6\,x_{1}x_{5}\longrightarrow }">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 6\,x_{1}x_{5}\longrightarrow }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/33d38bfde5c8ab9e56e9d15316db537abd3a8ac0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.768ex; height:2.509ex;" alt="{\displaystyle 6\,x_{1}x_{5}\longrightarrow }" loading="lazy">) 6 Möglichkeiten gibt, eine Menge mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n=6}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n=6}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f0365f0b9f2721ed3ebb488a96d7348d978acf8f.svg" 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=6}" loading="lazy"> Elementen zu <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k=2}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle k=2}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0bd301789e1f25a3da4be297ff637754ebee5f5d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.472ex; height:2.176ex;" alt="{\displaystyle k=2}" loading="lazy"> Partitionen mit 1 und 5 Elementen zu partitionieren,</dd>

<dd>(Monom <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 15\,x_{2}x_{4}\longrightarrow }">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 15\,x_{2}x_{4}\longrightarrow }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8597f120f30117b0b25a34a3ec66d222210faa58.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.931ex; height:2.509ex;" alt="{\displaystyle 15\,x_{2}x_{4}\longrightarrow }" loading="lazy">) 15 Möglichkeiten gibt, eine Menge mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n=6}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n=6}</annotation>

</semantics>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle k=2}</annotation>

</semantics>

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 10\,x_{3}^{2}\longrightarrow }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f9580030307adba9a91f3a0d5d42e6dfd3c09997.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:9.547ex; height:3.176ex;" alt="{\displaystyle 10\,x_{3}^{2}\longrightarrow }" loading="lazy">) 10 Möglichkeiten gibt, eine Menge mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n=6}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n=6}</annotation>

</semantics>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle k=2}</annotation>

</semantics>

<ul><li>Ein weiteres Beispiel ist</li></ul>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</msubsup>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle B_{6,3}(x_{1},x_{2},x_{3},x_{4})=15x_{1}^{2}x_{4}+60x_{1}x_{2}x_{3}+15x_{2}^{3}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/dc2a681231e93d35818ff9c38aa57a0d9929d6cb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:48.602ex; height:3.176ex;" alt="{\displaystyle B_{6,3}(x_{1},x_{2},x_{3},x_{4})=15x_{1}^{2}x_{4}+60x_{1}x_{2}x_{3}+15x_{2}^{3}}" loading="lazy"></dd></dl></dd>

<dd>da es

</mrow>

</mrow>

</msubsup>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 15\,x_{1}^{2}x_{4}\longrightarrow }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a938084a0f689929269ba3fabb1e7787bdc267f5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:11.931ex; height:3.176ex;" alt="{\displaystyle 15\,x_{1}^{2}x_{4}\longrightarrow }" loading="lazy">) 15 Möglichkeiten gibt, eine Menge mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n=6}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n=6}</annotation>

</semantics>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle k=3}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/662e06a2436f8a44fec791f5c794621f10dc8f30.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.472ex; height:2.176ex;" alt="{\displaystyle k=3}" loading="lazy"> Partitionen mit jeweils 1, 1 und 4 Elementen zu partitionieren,</dd>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 60\,x_{1}x_{2}x_{3}\longrightarrow }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bc94b86daa6a3c74cbfabfce6491491e7071635e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.315ex; height:2.509ex;" alt="{\displaystyle 60\,x_{1}x_{2}x_{3}\longrightarrow }" loading="lazy">) 60 Möglichkeiten gibt, eine Menge mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n=6}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n=6}</annotation>

</semantics>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle k=3}</annotation>

</semantics>

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 15\,x_{2}^{3}\longrightarrow }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c98da049baebb1b148007de196c8a39fb37209c2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:9.547ex; height:3.176ex;" alt="{\displaystyle 15\,x_{2}^{3}\longrightarrow }" loading="lazy">) 15 Möglichkeiten gibt, eine Menge mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n=6}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n=6}</annotation>

</semantics>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle k=3}</annotation>

</semantics>

<div class="mw-heading mw-heading2"><h2 id="Eigenschaften">Eigenschaften</h2></div>

<msub>

</mrow>

</msub>

<mrow>

</mrow>

</mfrac>

</mrow>

<mrow>

</mrow>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle B_{n,k}(1!,2!,\dots ,(n-k+1)!)={\binom {n}{k}}{\binom {n-1}{k-1}}(n-k)!}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/909c02d638e6bfdbf1876a385daebddd5b35fb33.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:52.477ex; height:6.176ex;" alt="{\displaystyle B_{n,k}(1!,2!,\dots ,(n-k+1)!)={\binom {n}{k}}{\binom {n-1}{k-1}}(n-k)!}" loading="lazy"></li></ul>

<div class="mw-heading mw-heading3"><h3 id="Bell-Polynome_und_Stirling-Zahlen">Bell-Polynome und Stirling-Zahlen</h3></div>

Der Wert des Bell-Polynoms <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B_{n,k}(x_{1},x_{2},\dots )}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle B_{n,k}(x_{1},x_{2},\dots )}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/91fe91d704b2db2c50f0723e4bb4f6cc4ea326d2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:15.665ex; height:3.009ex;" alt="{\displaystyle B_{n,k}(x_{1},x_{2},\dots )}" loading="lazy">, wenn alle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{i}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x_{i}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e87000dd6142b81d041896a30fe58f0c3acb2158.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.129ex; height:2.009ex;" alt="{\displaystyle x_{i}}" loading="lazy"> gleich 1 sind, ist eine <a href="Stirling-Zahl#Stirling-Zahlen_zweiter_Art" title="Stirling-Zahl">Stirling-Zahl zweiter Art</a>

<msub>

</mrow>

</msub>

<mrow>

</mfrac>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle B_{n,k}(1,1,\dots )=S(n,k)=\left\{{n \atop k}\right\}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4c553a25996032a96350566cf780fc9b48b69106.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:31.42ex; height:4.843ex;" alt="{\displaystyle B_{n,k}(1,1,\dots )=S(n,k)=\left\{{n \atop k}\right\}}" loading="lazy"> .</dd></dl>

Die Summe

</mrow>

</mrow>

</munderover>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</munderover>

<mrow>

</mfrac>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sum _{k=1}^{n}B_{n,k}(1,1,1,\dots )=\sum _{k=1}^{n}\left\{{n \atop k}\right\}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6f2c446a6973dee10f996afb47d353d45944751c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:31.053ex; height:6.843ex;" alt="{\displaystyle \sum _{k=1}^{n}B_{n,k}(1,1,1,\dots )=\sum _{k=1}^{n}\left\{{n \atop k}\right\}}" loading="lazy"></dd></dl>

entspricht der <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" 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}" loading="lazy">ten <a href="Bellsche_Zahl" title="Bellsche Zahl">Bellzahl</a>, welche die Anzahl der möglichen Partitionen einer Menge mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

</semantics>

<div class="mw-heading mw-heading3"><h3 id="Faltungsidentität">Faltungsidentität</h3></div>

Für Folgen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x_{n})_{n=1,2,\dotsc }}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (x_{n})_{n=1,2,\dotsc }}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0bfda2c86c4c40a114e961ba753ae32d76a9c073.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:11.339ex; height:3.009ex;" alt="{\displaystyle (x_{n})_{n=1,2,\dotsc }}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (y_{n})_{n=1,2,\dotsc }}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (y_{n})_{n=1,2,\dotsc }}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bbb6195f91f68b42654c93ed84c53973f7d4db2e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:11.148ex; height:3.009ex;" alt="{\displaystyle (y_{n})_{n=1,2,\dotsc }}" loading="lazy"> lässt sich eine Art <a href="Faltung_(Mathematik)" title="Faltung (Mathematik)">Faltung</a> definieren:

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</munderover>

<mrow>

</mrow>

</mfrac>

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (x\diamond y)_{n}=\sum _{j=1}^{n-1}{\binom {n}{j}}x_{j}y_{n-j}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/669a7514a582b12236d7231ecb618811b6a4bcb8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:25.917ex; height:7.676ex;" alt="{\displaystyle (x\diamond y)_{n}=\sum _{j=1}^{n-1}{\binom {n}{j}}x_{j}y_{n-j}}" loading="lazy"> ,</dd></dl>

wobei die Grenzen der Summe <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 1}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/92d98b82a3778f043108d4e20960a9193df57cbf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 1}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n-1}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n-1}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fbd0b0f32b28f51962943ee9ede4fb34198a2521.svg" 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}" loading="lazy"> anstelle von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 0}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2aae8864a3c1fec9585261791a809ddec1489950.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 0}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

</semantics>

Sei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{n}^{k\diamond }}">

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x_{n}^{k\diamond }}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/084858a783381acbc7e1ef6c7887201fe61041ac.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.24ex; height:2.843ex;" alt="{\displaystyle x_{n}^{k\diamond }}" loading="lazy"> der <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

</semantics>

<mrow>

</mrow>

</munder>

</mrow>

</mrow>

</munder>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \displaystyle \underbrace {x\diamond \cdots \diamond x} _{k\ \mathrm {Faktoren} }}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/87eb3cb854168dea1c39aea29151f66b19bfc7ec.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.838ex; margin-right: -0.028ex; width:9.8ex; height:5.176ex;" alt="{\displaystyle \displaystyle \underbrace {x\diamond \cdots \diamond x} _{k\ \mathrm {Faktoren} }}" loading="lazy"> ,</dd></dl>

dann gilt:

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<mfrac>

</mrow>

</mrow>

</msubsup>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle B_{n,k}(x_{1},\dots ,x_{n-k+1})={x_{n}^{k\diamond } \over k!}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/eacc28616b45100401cc0e980ef3c8b004f0e09d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:27.626ex; height:5.843ex;" alt="{\displaystyle B_{n,k}(x_{1},\dots ,x_{n-k+1})={x_{n}^{k\diamond } \over k!}}" loading="lazy"> .</dd></dl>

Die Faltungsidentität kann benutzt werden um einzelne Bell-Polynome zu berechnen. Die Berechnung von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B_{4,3}(x_{1},x_{2})}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle B_{4,3}(x_{1},x_{2})}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ca6620f437cdfc1fae7ecd5fd21e9f76cd95d5cf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:11.709ex; height:3.009ex;" alt="{\displaystyle B_{4,3}(x_{1},x_{2})}" loading="lazy"> ergibt sich mit

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x=(x_{1}\ ,\ x_{2}\ ,\ x_{3}\ ,\ x_{4}\ ,\dots )}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a7835992b71cb6146d765266b1a535dab038cd59.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.697ex; height:2.843ex;" alt="{\displaystyle x=(x_{1}\ ,\ x_{2}\ ,\ x_{3}\ ,\ x_{4}\ ,\dots )}" loading="lazy"></dd></dl>

</mrow>

</mrow>

</msubsup>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x\diamond x=(0,\ 2x_{1}^{2}\ ,\ 6x_{1}x_{2}\ ,\ 8x_{1}x_{3}+6x_{2}^{2}\ ,\dots )}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/63db3a55536723b11e581c93b86cf760e6cb6beb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:43.061ex; height:3.176ex;" alt="{\displaystyle x\diamond x=(0,\ 2x_{1}^{2}\ ,\ 6x_{1}x_{2}\ ,\ 8x_{1}x_{3}+6x_{2}^{2}\ ,\dots )}" loading="lazy"></dd></dl>

</mrow>

</mrow>

</msubsup>

</mrow>

</mrow>

</msubsup>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x\diamond x\diamond x=(0\ ,\ 0\ ,\ 6x_{1}^{3}\ ,\ 36x_{1}^{2}x_{2}\ ,\dots )}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7fa0a9d56da575378d36fb5b43f8168e54f23a85.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:37.174ex; height:3.176ex;" alt="{\displaystyle x\diamond x\diamond x=(0\ ,\ 0\ ,\ 6x_{1}^{3}\ ,\ 36x_{1}^{2}x_{2}\ ,\dots )}" loading="lazy"></dd></dl>

und dementsprechend,

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<mfrac>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

</mrow>

</msubsup>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle B_{4,3}(x_{1},x_{2})={\frac {(x\diamond x\diamond x)_{4}}{3!}}=6x_{1}^{2}x_{2}.}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3332d84e47f437f800ceba1a7c0068200bfd3c73.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:36.561ex; height:5.843ex;" alt="{\displaystyle B_{4,3}(x_{1},x_{2})={\frac {(x\diamond x\diamond x)_{4}}{3!}}=6x_{1}^{2}x_{2}.}" loading="lazy"></dd></dl>

<div class="mw-heading mw-heading2"><h2 id="Anwendungen">Anwendungen</h2></div>

<div class="mw-heading mw-heading3"><h3 id="Formel_von_Faà_di_Bruno">Formel von Faà di Bruno</h3></div>

<div class="hauptartikel" role="navigation">→ Hauptartikel: <a href="Formel_von_Fa%C3%A0_di_Bruno" title="Formel von Faà di Bruno">Formel von Faà di Bruno</a></div>

Die <a href="Formel_von_Fa%C3%A0_di_Bruno" title="Formel von Faà di Bruno">Formel von Faà di Bruno</a> kann mithilfe der Bell-Polynome wie folgt ausdrückt werden:

<mfrac>

<msup>

</mrow>

</msup>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

</mrow>

</mrow>

</munderover>

<msup>

</mrow>

</msup>

<msub>

</mrow>

</msub>

<mrow>

<mrow>

<msup>

</msup>

<msup>

</msup>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {d^{n} \over dx^{n}}f(g(x))=\sum _{k=0}^{n}f^{(k)}(g(x))B_{n,k}\left(g'(x),g''(x),\dots ,g^{(n-k+1)}(x)\right).}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ea83c08f3a6a28b816eca476ed8318a3d27e6627.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:64.568ex; height:7.009ex;" alt="{\displaystyle {d^{n} \over dx^{n}}f(g(x))=\sum _{k=0}^{n}f^{(k)}(g(x))B_{n,k}\left(g'(x),g''(x),\dots ,g^{(n-k+1)}(x)\right).}" loading="lazy"></dd></dl>

Auf ähnliche Art und Weise lässt sich eine <a href="Potenzreihe" title="Potenzreihe">Potenzreihen</a>-Version der Formel von Faà di Bruno aufstellen. Angenommen

</mrow>

</mrow>

</munderover>

<mfrac>

<msub>

</mrow>

</msub>

<mrow>

</mrow>

</mfrac>

</mrow>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</mrow>

</munderover>

<mfrac>

<msub>

</mrow>

</msub>

<mrow>

</mrow>

</mfrac>

</mrow>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(x)=\sum _{n=1}^{\infty }{a_{n} \over n!}x^{n}\qquad \mathrm {und} \qquad g(x)=\sum _{n=1}^{\infty }{b_{n} \over n!}x^{n}.}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/23f9024b76710e9ef60a9e2e524b9daa3b567c26.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:47.601ex; height:6.843ex;" alt="{\displaystyle f(x)=\sum _{n=1}^{\infty }{a_{n} \over n!}x^{n}\qquad \mathrm {und} \qquad g(x)=\sum _{n=1}^{\infty }{b_{n} \over n!}x^{n}.}" loading="lazy"></dd></dl>

Dann

</mrow>

</mrow>

</munderover>

<mfrac>

<mrow>

</mrow>

</mrow>

</munderover>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g(f(x))=\sum _{n=1}^{\infty }{\sum _{k=1}^{n}b_{k}B_{n,k}(a_{1},\dots ,a_{n-k+1}) \over n!}x^{n}.}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/513089423999e76c380395d1b7b547f129918c3f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:46.583ex; height:7.009ex;" alt="{\displaystyle g(f(x))=\sum _{n=1}^{\infty }{\sum _{k=1}^{n}b_{k}B_{n,k}(a_{1},\dots ,a_{n-k+1}) \over n!}x^{n}.}" loading="lazy"></dd></dl>

Die vollständigen Bell-Polynome tauchen in der Exponentialfunktion einer <a href="Formale_Potenzreihe" title="Formale Potenzreihe">formalen Potenzreihe</a> auf:

<mrow>

<mrow>

</mrow>

</mrow>

</munderover>

<mfrac>

<msub>

</mrow>

</msub>

<mrow>

</mrow>

</mfrac>

</mrow>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</mrow>

</mrow>

</munderover>

<mfrac>

<mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \exp \left(\sum _{n=1}^{\infty }{\frac {a_{n}}{n!}}x^{n}\right)=\sum _{n=0}^{\infty }{\frac {B_{n}(a_{1},\dots ,a_{n})}{n!}}x^{n}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/332dcd63e550265416f29c1bc2694d80bbda9eaf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:41.735ex; height:7.509ex;" alt="{\displaystyle \exp \left(\sum _{n=1}^{\infty }{\frac {a_{n}}{n!}}x^{n}\right)=\sum _{n=0}^{\infty }{\frac {B_{n}(a_{1},\dots ,a_{n})}{n!}}x^{n}}" loading="lazy"> .</dd></dl>

<div class="mw-heading mw-heading3"><h3 id="Momente_und_Kumulanten">Momente und Kumulanten</h3></div>

Die Summe

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</munderover>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle B_{n}(\kappa _{1},\dots ,\kappa _{n})=\sum _{k=1}^{n}B_{n,k}(\kappa _{1},\dots ,\kappa _{n-k+1})}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7eff29a8a62a91d2b11b8b78a590f860c5887d5c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:42.231ex; height:6.843ex;" alt="{\displaystyle B_{n}(\kappa _{1},\dots ,\kappa _{n})=\sum _{k=1}^{n}B_{n,k}(\kappa _{1},\dots ,\kappa _{n-k+1})}" loading="lazy"></dd></dl>

ist das <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" 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}" loading="lazy">te <a href="Moment_(Stochastik)" title="Moment (Stochastik)">Moment</a> einer <a href="Wahrscheinlichkeitsverteilung" class="mw-redirect" title="Wahrscheinlichkeitsverteilung">Wahrscheinlichkeitsverteilung</a>, deren erste <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" 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}" loading="lazy"> <a href="Kumulante" title="Kumulante">Kumulanten</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \kappa _{1},\dots ,\kappa _{n}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \kappa _{1},\dots ,\kappa _{n}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b7cfe4f9c18ec4c80f21ba0bdabb5caf13a799f2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.129ex; height:2.009ex;" alt="{\displaystyle \kappa _{1},\dots ,\kappa _{n}}" loading="lazy"> sind. Anders ausgedrückt ist das <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" 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}" loading="lazy">te Moment das <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" 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}" loading="lazy">te vollständige Bell-Polynom ausgewertet an den <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

</semantics>

<div class="mw-heading mw-heading3"><h3 id="Darstellung_von_Polynomfolgen_mit_binomialer_Eigenschaft">Darstellung von Polynomfolgen mit binomialer Eigenschaft</h3></div>

Für eine beliebige (skalare) Folge :<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{1},a_{2},a_{3},\dots }">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle a_{1},a_{2},a_{3},\dots }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b3b7eb2acf428a2c57ea6c3a379744fcce1d99c0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.677ex; height:2.009ex;" alt="{\displaystyle a_{1},a_{2},a_{3},\dots }" loading="lazy"> sei

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</munderover>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p_{n}(x)=\sum _{k=1}^{n}B_{n,k}(a_{1},\dots ,a_{n-k+1})x^{k}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/45f989859ff79de45e340d0301e6a51b44027de7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; margin-left: -0.089ex; width:35.127ex; height:6.843ex;" alt="{\displaystyle p_{n}(x)=\sum _{k=1}^{n}B_{n,k}(a_{1},\dots ,a_{n-k+1})x^{k}}" loading="lazy"> .</dd></dl>

Diese Polynomfolge erfüllt die binomiale Eigenschaft, d. h.

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</munderover>

<mrow>

</mrow>

</mfrac>

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p_{n}(x+y)=\sum _{k=0}^{n}{n \choose k}p_{k}(x)p_{n-k}(y)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/13b5052c2fa895d69387ad1fffc41150ea18ab7a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; margin-left: -0.089ex; width:34.153ex; height:7.009ex;" alt="{\displaystyle p_{n}(x+y)=\sum _{k=0}^{n}{n \choose k}p_{k}(x)p_{n-k}(y)}" loading="lazy"></dd></dl>

für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\geq 0}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n\geq 0}</annotation>

</semantics>

Für die <a href="Formale_Potenzreihe#Inverses_der_Komposition_(Umkehrfunktion)" title="Formale Potenzreihe">Inverse <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h^{-1}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle h^{-1}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8482555dae5cbaa401d8e0cb7d5eb4df203a81b5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.672ex; height:2.676ex;" alt="{\displaystyle h^{-1}}" loading="lazy"> der Komposition</a> der formalen Potenzreihe

</mrow>

</mrow>

</munderover>

<mfrac>

<msub>

</mrow>

</msub>

<mrow>

</mrow>

</mfrac>

</mrow>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle h(x)=\sum _{n=1}^{\infty }{a_{n} \over n!}x^{n}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ad67991b2ae95178eacf60cc909d5f272b175162.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:17.151ex; height:6.843ex;" alt="{\displaystyle h(x)=\sum _{n=1}^{\infty }{a_{n} \over n!}x^{n}}" loading="lazy"></dd></dl>

ergibt sich für alle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

</semantics>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</mrow>

</msubsup>

</mrow>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle h^{-1}{\bigl (}p_{n}^{\prime }(x){\bigr )}=np_{n-1}(x)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5daa2dcc34448f5c0e52ccc0a2158a18bda8e821.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:23.449ex; height:3.343ex;" alt="{\displaystyle h^{-1}{\bigl (}p_{n}^{\prime }(x){\bigr )}=np_{n-1}(x)}" loading="lazy"></dd></dl>

mit den obigen Polynomen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p_{n}(x)}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p_{n}(x)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1a47cca302b556e703b68e2a12168117208f47ff.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:5.616ex; height:2.843ex;" alt="{\displaystyle p_{n}(x)}" loading="lazy"> mit Koeffizienten in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{1},\dots ,a_{n-k+1}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle a_{1},\dots ,a_{n-k+1}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/77b13f6cbeb0a318a62fd9ae71284dcd9454a941.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.146ex; height:2.009ex;" alt="{\displaystyle a_{1},\dots ,a_{n-k+1}}" loading="lazy"> .

<div class="mw-heading mw-heading2"><h2 id="Literatur">Literatur</h2></div>

<ul><li><a href="Eric_Temple_Bell" class="mw-redirect" title="Eric Temple Bell">Eric Temple Bell</a>: <cite style="font-style:italic">Partition Polynomials</cite>. In: <cite style="font-style:italic"><a href="Annals_of_Mathematics" title="Annals of Mathematics">Annals of Mathematics</a></cite>. Band 29, Nr. 1/4, 1927, S. 38–46, <a href="Digital_Object_Identifier" title="Digital Object Identifier">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307/1967979">10.2307/1967979</a>, <a href="JSTOR" title="JSTOR">JSTOR</a>:<a rel="nofollow" class="external text" href="http://www.jstor.org/stable/1967979">1967979</a>.<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rfr_id=info:sid/de.wikipedia.org:Bell-Polynom&rft.atitle=Partition+Polynomials&rft.au=Eric+Temple+Bell&rft.date=1927&rft.doi=10.2307%2F1967979&rft.genre=journal&rft.issue=1%2F4&rft.jtitle=Annals+of+Mathematics&rft.pages=38-46&rft.volume=29" style="display:none"> </li>

<li>Louis Comtet: <cite style="font-style:italic">Advanced Combinatorics: The Art of Finite and Infinite Expansions</cite>. Reidel Publishing Company, Dordrecht, Holland / Boston, U.S. 1974.<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Bell-Polynom&rft.au=Louis+Comtet&rft.btitle=Advanced+Combinatorics%3A+The+Art+of+Finite+and+Infinite+Expansions&rft.date=1974&rft.genre=book&rft.place=Dordrecht%2C+Holland+%2F+Boston%2C+U.S.&rft.pub=Reidel+Publishing+Company" style="display:none"> </li>

<li><a href="Steven_Roman" title="Steven Roman">Steven Roman</a>: <cite style="font-style:italic">The Umbral Calculus</cite>. Academic Press, 1984, ISBN 0-08-087430-4 (<a rel="nofollow" class="external text" href="https://www.123library.org/ebook/id/41593/">123library.org</a>).<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Bell-Polynom&rft.au=Steven+Roman&rft.btitle=The+Umbral+Calculus&rft.date=1984&rft.genre=book&rft.isbn=0080874304&rft.pub=Academic+Press" style="display:none"> </li>

<li>Vassily G. Voinov, Mikhail S. Nikulin: <cite style="font-style:italic">On power series, Bell polynomials, Hardy-Ramanujan-Rademacher problem and its statistical applications</cite>. In: <cite style="font-style:italic">Kybernetika</cite>. Band 30, Nr. 3, 1994, <a href="Internationale_Standardnummer_f%C3%BCr_fortlaufende_Sammelwerke" title="Internationale Standardnummer für fortlaufende Sammelwerke">ISSN</a> <a rel="nofollow" class="external text" href="https://zdb-katalog.de/list.xhtml?t=iss%3D%220023-5954%22&key=cql">0023-5954</a>, S. 343–358 (<a rel="nofollow" class="external text" href="https://www.kybernetika.cz/content/1994/3/343/paper.pdf">kybernetika.cz</a> [PDF]).<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rfr_id=info:sid/de.wikipedia.org:Bell-Polynom&rft.atitle=On+power+series%2C+Bell+polynomials%2C+Hardy-Ramanujan-Rademacher+problem+and+its+statistical+applications&rft.au=Vassily+G.+Voinov%2C+Mikhail+S.+Nikulin&rft.date=1994&rft.genre=journal&rft.issn=0023-5954&rft.issue=3&rft.jtitle=Kybernetika&rft.pages=343-358&rft.volume=30" style="display:none"> </li>

<li><a href="George_Andrews_(Mathematiker)" title="George Andrews (Mathematiker)">George Andrews</a>: <cite style="font-style:italic">The Theory of Partitions</cite> (= <cite style="font-style:italic">Cambridge Mathematical Library</cite>). 1. Auflage. Cambridge University Press, 1998, ISBN 0-521-63766-X, S. 204–211.<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Bell-Polynom&rft.au=George+Andrews&rft.btitle=The+Theory+of+Partitions&rft.date=1998&rft.edition=1.&rft.genre=book&rft.isbn=052163766X&rft.pages=204-211&rft.pub=Cambridge+University+Press&rft.series=Cambridge+Mathematical+Library" style="display:none"> </li>

<li>Silvia Noschese, Paolo E. Ricci: <cite style="font-style:italic">Differentiation of Multivariable Composite Functions and Bell Polynomials</cite>. In: <cite style="font-style:italic">Journal of Computational Analysis and Applications</cite>. Band 5, Nr. 3, 2003, S. 333–340, <a href="Digital_Object_Identifier" title="Digital Object Identifier">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1023/A%3A1023227705558">10.1023/A:1023227705558</a>.<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rfr_id=info:sid/de.wikipedia.org:Bell-Polynom&rft.atitle=Differentiation+of+Multivariable+Composite+Functions+and+Bell+Polynomials&rft.au=Silvia+Noschese%2C+Paolo+E.+Ricci&rft.date=2003&rft.doi=10.1023%2FA%3A1023227705558&rft.genre=journal&rft.issue=3&rft.jtitle=Journal+of+Computational+Analysis+and+Applications&rft.pages=333-340&rft.volume=5" style="display:none"> </li>

<li>Moncef Abbas, Sadek Bouroubi: <cite style="font-style:italic">On new identities for Bell’s polynomial</cite>. In: <cite style="font-style:italic">Disc. Math</cite>. Band 293, 2005, S. 5–10, <a href="Digital_Object_Identifier" title="Digital Object Identifier">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016/j.disc.2004.08.023">10.1016/j.disc.2004.08.023</a>.<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Bell-Polynom&rft.atitle=On+new+identities+for+Bell%E2%80%99s+polynomial&rft.au=Moncef+Abbas%2C+Sadek+Bouroubi&rft.btitle=Disc.+Math&rft.date=2005&rft.doi=10.1016%2Fj.disc.2004.08.023&rft.genre=book&rft.pages=5-10&rft.volume=293" style="display:none"> </li>

<li>Khristo N. Boyadzhiev: <cite style="font-style:italic">Exponential Polynomials, Stirling Numbers, and Evaluation of Some Gamma Integrals</cite>. In: <cite style="font-style:italic">Abstract and Applied Analysis</cite>. 2009, <a href="Digital_Object_Identifier" title="Digital Object Identifier">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1155/2009%2F168672">10.1155/2009/168672</a>.<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Bell-Polynom&rft.atitle=Exponential+Polynomials%2C+Stirling+Numbers%2C+and+Evaluation+of+Some+Gamma+Integrals&rft.au=Khristo+N.+Boyadzhiev&rft.btitle=Abstract+and+Applied+Analysis&rft.date=2009&rft.doi=10.1155%2F2009%2F168672&rft.genre=book" style="display:none"> </li>

<li>Martin Griffiths: <cite style="font-style:italic">Families of sequences from a class of multinomial sums</cite>. In: <cite style="font-style:italic">Journal of Integer Sequences</cite>. Band 15, 2012 (<a rel="nofollow" class="external text" href="https://cs.uwaterloo.ca/journals/JIS/VOL15/Griffiths/griffiths20.html">cs.uwaterloo.ca</a>).<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Bell-Polynom&rft.atitle=Families+of+sequences+from+a+class+of+multinomial+sums&rft.au=Martin+Griffiths&rft.btitle=Journal+of+Integer+Sequences&rft.date=2012&rft.genre=book&rft.volume=15" style="display:none"> </li></ul>

<div class="mw-heading mw-heading2"><h2 id="Einzelnachweise">Einzelnachweise</h2></div>

<li id="cite_note-1"><a href="#cite_ref-1">↑</a> Nikita Alexeev, Anna Pologova, Max A. Alekseyev: <cite style="font-style:italic">Generalized Hultman Numbers and Cycle Structures of Breakpoint Graphs</cite>. In: <cite style="font-style:italic">Journal of Computational Biology</cite>. 24. Jahrgang, Nr. 2, 2017, S. 93–105, <a href="Digital_Object_Identifier" title="Digital Object Identifier">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1089/cmb.2016.0190">10.1089/cmb.2016.0190</a>, <a href="ArXiv" title="ArXiv">arxiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/1503.05285">1503.05285</a>.<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rfr_id=info:sid/de.wikipedia.org:Bell-Polynom&rft.atitle=Generalized+Hultman+Numbers+and+Cycle+Structures+of+Breakpoint+Graphs&rft.au=Nikita%26%2332%3BAlexeev%2C%26%2332%3BAnna+Pologova%2C+Max+A.+Alekseyev&rft.date=2017&rft.doi=10.1089%2Fcmb.2016.0190&rft.genre=journal&rft.issue=2&rft.jtitle=Journal+of+Computational+Biology&rft.pages=93-105&rft.volume=24.+Jahrgang" style="display:none"> , sect. 4.2

</li>

</ol></div><div><div class="zim-footer">

Dieser Artikel wurde von <a class="external text" title="Zuletzt bearbeitet am 2023-05-01" href="https://de.wikipedia.org/wiki/?title=Bell-Polynom&oldid=233348045">Wikipedia</a> herausgegeben. Der Text ist unter <a class="external text" href="https://creativecommons.org/licenses/by-sa/4.0/deed.de">Creative Commons Attribution-Share Alike 4.0</a> verfügbar, sofern nicht anders angegeben. Für die Mediendateien können zusätzliche Bedingungen gelten.

</div>

</div></div>

</div>

</main>

</div>

</body></html>