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.

1, 1, 3, 9, 37, 177, 1054, 7237, 57447, 512664, 5101453, 55870885, 668438484, 8667987140, 121123281293, 1814038728900, 28988885491655, 492308367375189, 8854101716492463, 168108959387012804, 3360171602215686668, 70527588239926854144, 1550926052235372201700

Offset: 0

Author

Gus Wiseman, Nov 18 2018

Keywords

• nonn

Examples

			The a(3) = 9 matrices:
  [3]
.
  [2 0] [1 1]
  [0 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]
		

Crossrefs

Cf. A000700, A006052, A007016, A007716, A120732, A319056, A319616, A320451, A321719, A321722, A321732, A321735, A321736, A321739.

Programs

Mathematica

prs2mat[prs_]:=Table[Count[prs,{i,j}],{i,Union[First/@prs]},{j,Union[Last/@prs]}];
multsubs[set_,k_]:=If[k==0,{{}},Join@@Table[Prepend[#,set[[i]]]&/@multsubs[Drop[set,i-1],k-1],{i,Length[set]}]];
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}]

Formula

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).

Extensions

a(11) - a(22) from Ludovic Schwob, Sep 29 2023