cp's OEIS Frontend

A321734 Number of nonnegative integer square matrices with sum of entries equal to n, no zero rows or columns, weakly decreasing row and column sums, and the same row sums as column sums.

%I A321734 #10 Sep 29 2023 08:27:24
%S A321734 1,1,3,9,37,177,1054,7237,57447,512664,5101453,55870885,668438484,
%T A321734 8667987140,121123281293,1814038728900,28988885491655,492308367375189,
%U A321734 8854101716492463,168108959387012804,3360171602215686668,70527588239926854144,1550926052235372201700
%N A321734 Number of nonnegative integer square matrices with sum of entries equal to n, no zero rows or columns, weakly decreasing row and column sums, and the same row sums as column sums.
%F A321734 Let c(y) be the coefficient of m(y) in h(y), where m is monomial symmetric functions and h is homogeneous symmetric functions. Then a(n) = Sum_{|y| = n} c(y).
%e A321734 The a(3) = 9 matrices:
%e A321734   [3]
%e A321734 .
%e A321734   [2 0] [1 1]
%e A321734   [0 1] [1 0]
%e A321734 .
%e A321734   [1 0 0] [1 0 0] [0 1 0] [0 1 0] [0 0 1] [0 0 1]
%e A321734   [0 1 0] [0 0 1] [1 0 0] [0 0 1] [1 0 0] [0 1 0]
%e A321734   [0 0 1] [0 1 0] [0 0 1] [1 0 0] [0 1 0] [1 0 0]
%t A321734 prs2mat[prs_]:=Table[Count[prs,{i,j}],{i,Union[First/@prs]},{j,Union[Last/@prs]}];
%t A321734 multsubs[set_,k_]:=If[k==0,{{}},Join@@Table[Prepend[#,set[[i]]]&/@multsubs[Drop[set,i-1],k-1],{i,Length[set]}]];
%t A321734 Table[Length[Select[multsubs[Tuples[Range[n],2],n],And[Union[First/@#]==Range[Max@@First/@#]==Union[Last/@#],OrderedQ[Total/@prs2mat[#]],OrderedQ[Total/@Transpose[prs2mat[#]]],Total/@prs2mat[#]==Total/@Transpose[prs2mat[#]]]&]],{n,5}]
%Y A321734 Cf. A000700, A006052, A007016, A007716, A120732, A319056, A319616, A320451, A321719, A321722, A321732, A321735, A321736, A321739.
%K A321734 nonn
%O A321734 0,3
%A A321734 _Gus Wiseman_, Nov 18 2018
%E A321734 a(11) - a(22) from _Ludovic Schwob_, Sep 29 2023