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

0,3

Links

Table of n, a(n) for n=0..22.

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).

Example

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]

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}]

Crossrefs

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

Sequence in context: A366433 A319119 A006229 * A008986 A105215 A158053

Adjacent sequences: A321731 A321732 A321733 * A321735 A321736 A321737

Keyword

nonn

Author

Gus Wiseman, Nov 18 2018

Extensions

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

Status

approved