StirlingS2[n, m] == Sum[\[Ellipsis] Sum[KroneckerDelta[m, Sum[Subscript[k, j], {j, 1, n}]] KroneckerDelta[n, Sum[j Subscript[k, j], {j, 1, n}]] (n!/Product[j!^Subscript[k, j] Subscript[k, j]!, {j, 1, n}]), {Subscript[k, n], 0, Max[m, n]}], {Subscript[k, 1], 0, Max[m, n]}, {Subscript[k, 2], 0, Max[m, n]}]

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<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <msubsup> <semantics> <mi> &#119982; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[ScriptCapitalS]&quot;, StirlingS2] </annotation> </semantics> <mi> n </mi> <mrow> <mo> ( </mo> <mi> m </mi> <mo> ) </mo> </mrow> </msubsup> <mo> &#10869; </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <msub> <mi> k </mi> <mn> 1 </mn> </msub> <mo> = </mo> <mn> 0 </mn> </mrow> <mrow> <mi> max </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> m </mi> <mo> , </mo> <mi> n </mi> </mrow> <mo> ) </mo> </mrow> </munderover> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <msub> <mi> k </mi> <mn> 2 </mn> </msub> <mo> = </mo> <mn> 0 </mn> </mrow> <mrow> <mi> max </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> m </mi> <mo> , </mo> <mi> n </mi> </mrow> <mo> ) </mo> </mrow> </munderover> <mrow> <mo> &#8230; </mo> <mo> &#8290; </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <msub> <mi> k </mi> <mi> n </mi> </msub> <mo> = </mo> <mn> 0 </mn> </mrow> <mrow> <mi> max </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> m </mi> <mo> , </mo> <mi> n </mi> </mrow> <mo> ) </mo> </mrow> </munderover> <mrow> <mfrac> <mrow> <mtext> </mtext> <mrow> <mi> n </mi> <mo> ! </mo> </mrow> </mrow> <mrow> <munderover> <mo> &#8719; </mo> <mrow> <mi> j </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> n </mi> </munderover> <mrow> <msup> <mrow> <mi> j </mi> <mo> ! </mo> </mrow> <msub> <mi> k </mi> <mi> j </mi> </msub> </msup> <mo> &#8290; </mo> <mrow> <msub> <mi> k </mi> <mi> j </mi> </msub> <mo> ! </mo> </mrow> </mrow> </mrow> </mfrac> <mo> &#8290; </mo> <msub> <semantics> <mi> &#948; </mi> <annotation-xml encoding='MathML-Content'> <ci> KroneckerDelta </ci> </annotation-xml> </semantics> <mrow> <mi> m </mi> <mo> , </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> j </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> n </mi> </munderover> <msub> <mi> k </mi> <mi> j </mi> </msub> </mrow> </mrow> </msub> <mo> &#8290; </mo> <msub> <semantics> <mi> &#948; </mi> <annotation-xml encoding='MathML-Content'> <ci> KroneckerDelta </ci> </annotation-xml> </semantics> <mrow> <mi> n </mi> <mo> , </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> j </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> n </mi> </munderover> <mrow> <mi> j </mi> <mo> &#8290; </mo> <msub> <mi> k </mi> <mi> j </mi> </msub> </mrow> </mrow> </mrow> </msub> </mrow> </mrow> </mrow> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <ci> StirlingS2 </ci> <ci> n </ci> <ci> m </ci> </apply> <apply> <sum /> <bvar> <apply> <ci> Subscript </ci> <ci> k </ci> <cn type='integer'> 2 </cn> </apply> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <apply> <max /> <ci> m </ci> <ci> n </ci> </apply> </uplimit> <apply> <sum /> <bvar> <apply> <ci> Subscript </ci> <ci> k </ci> <cn type='integer'> 1 </cn> </apply> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <apply> <max /> <ci> m </ci> <ci> n </ci> </apply> </uplimit> <apply> <times /> <ci> &#8230; </ci> <apply> <sum /> <bvar> <apply> <ci> Subscript </ci> <ci> k </ci> <ci> n </ci> </apply> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <apply> <max /> <ci> m </ci> <ci> n </ci> </apply> </uplimit> <apply> <times /> <apply> <times /> <apply> <factorial /> <ci> n </ci> </apply> <apply> <power /> <apply> <product /> <bvar> <ci> j </ci> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <ci> n </ci> </uplimit> <apply> <times /> <apply> <power /> <apply> <factorial /> <ci> j </ci> </apply> <apply> <ci> Subscript </ci> <ci> k </ci> <ci> j </ci> </apply> </apply> <apply> <factorial /> <apply> <ci> Subscript </ci> <ci> k </ci> <ci> j </ci> </apply> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> KroneckerDelta </ci> <ci> m </ci> <apply> <sum /> <bvar> <ci> j </ci> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <ci> n </ci> </uplimit> <apply> <ci> Subscript </ci> <ci> k </ci> <ci> j </ci> </apply> </apply> </apply> <apply> <ci> KroneckerDelta </ci> <ci> n </ci> <apply> <sum /> <bvar> <ci> j </ci> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <ci> n </ci> </uplimit> <apply> <times /> <ci> j </ci> <apply> <ci> Subscript </ci> <ci> k </ci> <ci> j </ci> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

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2001-10-29