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A321735
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.
6
1, 1, 2, 7, 30, 153, 939, 6653, 53743, 486576
OFFSET
0,3
FORMULA
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).
EXAMPLE
The a(3) = 7 matrices:
[1 1]
[1 0]
.
[1 0 0] [1 0 0] [0 1 0] [0 1 0] [0 0 1] [0 0 1]
[0 1 0] [0 0 1] [1 0 0] [0 0 1] [1 0 0] [0 1 0]
[0 0 1] [0 1 0] [0 0 1] [1 0 0] [0 1 0] [1 0 0]
MATHEMATICA
prs2mat[prs_]:=Table[Count[prs, {i, j}], {i, Union[First/@prs]}, {j, Union[Last/@prs]}];
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}]
KEYWORD
nonn,more
AUTHOR
Gus Wiseman, Nov 18 2018
STATUS
approved