Sum[Sum[Product[1/Subscript[k,i,j]!,{i,1,m},{j,1,m}],{Subscript[k,m,n],0, o}],…, {Subscript[k,1,1],0,o}] p!/Product[ Subscript[a,i]!*Product[Subscript[b,j]!,{j,1,n}],{i,1, m}]/;(Sum[Subscript[k,i,j],{j,1,n}]Subscript[a,i]&& Sum[Subscript[k,i,j],{i,1,m}]Subscript[b,i]&& Sum[Subscript[a,i],{i,1,m}]Sum[Subscript[b,j],{j,1,n}] p&&o=Max[Subscript[k,1,1],…,Subscript[k,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> <mrow> <mrow> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <msub> <mi> k </mi> <mrow> <mn> 1 </mn> <mo> , </mo> <mn> 1 </mn> </mrow> </msub> <mo> = </mo> <mn> 0 </mn> </mrow> <mi> o </mi> </munderover> <mrow> <mo> &#8230; </mo> <mo> &#8290; </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <msub> <mi> k </mi> <mrow> <mi> m </mi> <mo> , </mo> <mi> n </mi> </mrow> </msub> <mo> = </mo> <mn> 0 </mn> </mrow> <mi> o </mi> </munderover> <mrow> <munderover> <mo> &#8719; </mo> <mrow> <mi> i </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> m </mi> </munderover> <mrow> <munderover> <mo> &#8719; </mo> <mrow> <mi> j </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> m </mi> </munderover> <mfrac> <mn> 1 </mn> <mrow> <msub> <mi> k </mi> <mrow> <mi> i </mi> <mo> , </mo> <mi> j </mi> </mrow> </msub> <mo> ! </mo> </mrow> </mfrac> </mrow> </mrow> </mrow> </mrow> </mrow> <mo> &#10869; </mo> <mfrac> <mrow> <mi> p </mi> <mo> ! </mo> </mrow> <mrow> <munderover> <mo> &#8719; </mo> <mrow> <mi> i </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> m </mi> </munderover> <mrow> <mrow> <msub> <mi> a </mi> <mi> i </mi> </msub> <mo> ! </mo> </mrow> <mo> &#8290; </mo> <mrow> <munderover> <mo> &#8719; </mo> <mrow> <mi> j </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> n </mi> </munderover> <mrow> <msub> <mi> b </mi> <mi> j </mi> </msub> <mo> ! </mo> </mrow> </mrow> </mrow> </mrow> </mfrac> </mrow> <mo> /; </mo> <mrow> <mrow> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> j </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> n </mi> </munderover> <msub> <mi> k </mi> <mrow> <mi> i </mi> <mo> , </mo> <mi> j </mi> </mrow> </msub> </mrow> <mo> &#10869; </mo> <msub> <mi> a </mi> <mi> i </mi> </msub> </mrow> <mo> &#8743; </mo> <mrow> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> i </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> m </mi> </munderover> <msub> <mi> k </mi> <mrow> <mi> i </mi> <mo> , </mo> <mi> j </mi> </mrow> </msub> </mrow> <mo> &#10869; </mo> <msub> <mi> b </mi> <mi> i </mi> </msub> </mrow> <mo> &#8743; </mo> <mrow> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> i </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> m </mi> </munderover> <msub> <mi> a </mi> <mi> i </mi> </msub> </mrow> <mo> &#10869; </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> j </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> n </mi> </munderover> <msub> <mi> b </mi> <mi> j </mi> </msub> </mrow> <mo> &#10869; </mo> <mi> p </mi> </mrow> <mo> &#8743; </mo> <mi> o </mi> </mrow> </mrow> <mo> = </mo> <mrow> <mi> Max </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <msub> <mi> k </mi> <mrow> <mn> 1 </mn> <mo> , </mo> <mn> 1 </mn> </mrow> </msub> <mo> , </mo> <mo> &#8230; </mo> <mo> , </mo> <msub> <mi> k </mi> <mrow> <mi> m </mi> <mo> , </mo> <mi> n </mi> </mrow> </msub> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Set </ci> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <sum /> <bvar> <apply> <ci> Subscript </ci> <ci> k </ci> <cn type='integer'> 1 </cn> <cn type='integer'> 1 </cn> </apply> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <ci> o </ci> </uplimit> <apply> <times /> <ci> &#8230; </ci> <apply> <sum /> <bvar> <apply> <ci> Subscript </ci> <ci> k </ci> <ci> m </ci> <ci> n </ci> </apply> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <ci> o </ci> </uplimit> <apply> <product /> <bvar> <ci> j </ci> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <ci> m </ci> </uplimit> <apply> <product /> <bvar> <ci> i </ci> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <ci> m </ci> </uplimit> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <factorial /> <apply> <ci> Subscript </ci> <ci> k </ci> <ci> i </ci> <ci> j </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <factorial /> <ci> p </ci> </apply> <apply> <power /> <apply> <product /> <bvar> <ci> i </ci> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <ci> m </ci> </uplimit> <apply> <times /> <apply> <factorial /> <apply> <ci> Subscript </ci> <ci> a </ci> <ci> i </ci> </apply> </apply> <apply> <product /> <bvar> <ci> j </ci> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <ci> n </ci> </uplimit> <apply> <factorial /> <apply> <ci> Subscript </ci> <ci> b </ci> <ci> j </ci> </apply> </apply> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <and /> <apply> <eq /> <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> i </ci> <ci> j </ci> </apply> </apply> <apply> <ci> Subscript </ci> <ci> a </ci> <ci> i </ci> </apply> </apply> <apply> <eq /> <apply> <sum /> <bvar> <ci> i </ci> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <ci> m </ci> </uplimit> <apply> <ci> Subscript </ci> <ci> k </ci> <ci> i </ci> <ci> j </ci> </apply> </apply> <apply> <ci> Subscript </ci> <ci> b </ci> <ci> i </ci> </apply> </apply> <apply> <eq /> <apply> <sum /> <bvar> <ci> i </ci> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <ci> m </ci> </uplimit> <apply> <ci> Subscript </ci> <ci> a </ci> <ci> i </ci> </apply> </apply> <apply> <sum /> <bvar> <ci> j </ci> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <ci> n </ci> </uplimit> <apply> <ci> Subscript </ci> <ci> b </ci> <ci> j </ci> </apply> </apply> <ci> p </ci> </apply> <ci> o </ci> </apply> </apply> <apply> <max /> <apply> <ci> Subscript </ci> <ci> k </ci> <cn type='integer'> 1 </cn> <cn type='integer'> 1 </cn> </apply> <ci> &#8230; </ci> <apply> <ci> Subscript </ci> <ci> k </ci> <ci> m </ci> <ci> n </ci> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

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2002-12-18