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A321652 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.

%I A321652 #18 Jun 11 2025 17:19:17
%S A321652 1,1,5,19,107,573,4050,29093,249301,2271020,23378901,257871081,
%T A321652 3132494380,40693204728,572089068459,8566311524788,137165829681775,
%U A321652 2327192535461323,41865158805428687,793982154675640340,15863206077534914434,332606431999260837036,7309310804287502958322,167896287022455809865568
%N A321652 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 A321652 Ludovic Schwob, <a href="/A321652/b321652.txt">Table of n, a(n) for n = 0..39</a>
%H A321652 Ludovic Schwob, <a href="https://arxiv.org/abs/2506.04007">On the enumeration of double cosets and self-inverse double cosets</a>, arXiv:2506.04007 [math.CO], 2025. See p. 13.
%F A321652 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 A321652 The a(3) = 19 matrices:
%e A321652   [3] [2 1] [1 1 1]
%e A321652 .
%e A321652   [2] [2 0] [1 1] [1 1 0] [1 0 1] [0 1 1]
%e A321652   [1] [0 1] [1 0] [0 0 1] [0 1 0] [1 0 0]
%e A321652 .
%e A321652   [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 A321652   [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 A321652   [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 A321652 prs2mat[prs_]:=Table[Count[prs,{i,j}],{i,Union[First/@prs]},{j,Union[Last/@prs]}];
%t A321652 multsubs[set_,k_]:=If[k==0,{{}},Join@@Table[Prepend[#,set[[i]]]&/@multsubs[Drop[set,i-1],k-1],{i,Length[set]}]];
%t A321652 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 A321652 Cf. A000219, A001970, A007716, A068313, A114736, A120733, A319646, A321645, A321653, A321654, A321655.
%K A321652 nonn
%O A321652 0,3
%A A321652 _Gus Wiseman_, Nov 15 2018
%E A321652 a(10) onwards from _Ludovic Schwob_, Aug 29 2023