%I #4 Nov 19 2018 07:22:23
%S 1,1,2,7,30,153,939,6653,53743,486576
%N Number of (0,1)-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 Let c(y) be the coefficient of m(y) in e(y), where m is monomial symmetric functions and e is elementary symmetric functions. Then a(n) = Sum_{|y| = n} c(y).
%e The a(3) = 7 matrices:
%e [1 1]
%e [1 0]
%e .
%e [1 0 0] [1 0 0] [0 1 0] [0 1 0] [0 0 1] [0 0 1]
%e [0 1 0] [0 0 1] [1 0 0] [0 0 1] [1 0 0] [0 1 0]
%e [0 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 Table[Length[Select[Subsets[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 Cf. A000700, A007016, A049311, A054976, A057151, A104602, A320451, A321719, A321723, A321732, A321733, A321736, A321739.
%K nonn,more
%O 0,3
%A _Gus Wiseman_, Nov 18 2018
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