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%I #14 Aug 29 2023 15:35:28
%S 1,1,5,19,107,573,4050,29093,249301,2271020,23378901,257871081,
%T 3132494380,40693204728,572089068459,8566311524788,137165829681775,
%U 2327192535461323,41865158805428687,793982154675640340,15863206077534914434,332606431999260837036,7309310804287502958322,167896287022455809865568
%N Number of nonnegative integer matrices with sum of entries equal to n and no zero rows or columns, with weakly decreasing row sums and column sums.
%H Ludovic Schwob, <a href="/A321652/b321652.txt">Table of n, a(n) for n = 0..39</a>
%F Sum of coefficients in the expansions of all homogeneous symmetric functions in terms of monomial symmetric functions. In other words, if Sum_{|y| = n} h(y) = Sum_{|y| = n} c_y * m(y), then a(n) = Sum_{|y| = n} c_y.
%e The a(3) = 19 matrices:
%e [3] [2 1] [1 1 1]
%e .
%e [2] [2 0] [1 1] [1 1 0] [1 0 1] [0 1 1]
%e [1] [0 1] [1 0] [0 0 1] [0 1 0] [1 0 0]
%e .
%e [1] [1 0] [1 0] [1 0 0] [1 0 0] [0 1] [0 1 0] [0 1 0] [0 0 1] [0 0 1]
%e [1] [1 0] [0 1] [0 1 0] [0 0 1] [1 0] [1 0 0] [0 0 1] [1 0 0] [0 1 0]
%e [1] [0 1] [1 0] [0 0 1] [0 1 0] [1 0] [0 0 1] [1 0 0] [0 1 0] [1 0 0]
%t prs2mat[prs_]:=Table[Count[prs,{i,j}],{i,Union[First/@prs]},{j,Union[Last/@prs]}];
%t multsubs[set_,k_]:=If[k==0,{{}},Join@@Table[Prepend[#,set[[i]]]&/@multsubs[Drop[set,i-1],k-1],{i,Length[set]}]];
%t Table[Length[Select[multsubs[Tuples[Range[n],2],n],And[Union[First/@#]==Range[Max@@First/@#],Union[Last/@#]==Range[Max@@Last/@#],OrderedQ[Total/@prs2mat[#]],OrderedQ[Total/@Transpose[prs2mat[#]]]]&]],{n,6}]
%Y Cf. A000219, A001970, A007716, A068313, A114736, A120733, A319646, A321645, A321653, A321654, A321655.
%K nonn
%O 0,3
%A _Gus Wiseman_, Nov 15 2018
%E a(10) onwards from _Ludovic Schwob_, Aug 29 2023
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