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.

1, 1, 2, 7, 30, 153, 939, 6653, 53743, 486576

Offset

0,3

Links

Table of n, a(n) for n=0..9.

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}]

Crossrefs

Cf. A000700, A007016, A049311, A054976, A057151, A104602, A320451, A321719, A321723, A321732, A321733, A321736, A321739.

Sequence in context: A030896 A209801 A030848 * A030975 A144495 A154259

Adjacent sequences: A321732 A321733 A321734 * A321736 A321737 A321738

Keyword

nonn,more

Author

Gus Wiseman, Nov 18 2018

Status

approved