A068313 Number of (0,1)-matrices with sum of entries equal to n and no zero rows or columns, with weakly decreasing row sums and column sums.

1, 4, 15, 82, 457, 3231, 24055, 209375, 1955288, 20455936, 229830841, 2828166755, 37228913365, 528635368980, 7990596990430, 128909374528433, 2202090635802581, 39837079499488151, 759320365206705013, 15234890522990662422, 320634889654149218205, 7068984425261215971205

Offset: 1

Author

Axel Kohnert (axel.kohnert(AT)uni-bayreuth.de), Feb 25 2002

Keywords

• nonn

Comments

This is the sum over the matrix of base change from the elementary symmetric functions to the monomial symmetric functions.

Examples

			a(2) = 4 because there are 4 different 0-1 matrices of weight 2: 1 10 01 11,1, 01, 10.
From _Gus Wiseman_, Nov 15 2018: (Start)
The a(3) = 15 matrices:
  [1 1 1]
.
  [1 1] [1 1 0] [1 0 1] [0 1 1]
  [1 0] [0 0 1] [0 1 0] [1 0 0]
.
  [1] [1 0] [1 0] [1 0 0] [1 0 0] [0 1] [0 1 0] [0 1 0] [0 0 1] [0 0 1]
  [1] [1 0] [0 1] [0 1 0] [0 0 1] [1 0] [1 0 0] [0 0 1] [1 0 0] [0 1 0]
  [1] [0 1] [1 0] [0 0 1] [0 1 0] [1 0] [0 0 1] [1 0 0] [0 1 0] [1 0 0]
(End)
		

References

• I. G. Macdonald, Symmetric Functions and Hall Polynomials, Oxford 1979, p. 57.

Links

• Ludovic Schwob, Table of n, a(n) for n = 1..37
• Ludovic Schwob, On the enumeration of double cosets and self-inverse double cosets, arXiv:2506.04007 [math.CO], 2025. See p. 14.

Crossrefs

Cf. A000219, A001970, A007716, A049311, A101370, A117433, A120733, A321646, A321652, A321653, A321654.

Programs

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/@#]==Range[Max@@Last/@#],OrderedQ[Total/@prs2mat[#]],OrderedQ[Total/@T[prs2mat[#]]]]&]],{n,5}] (* Gus Wiseman, Nov 15 2018 *)

Extensions

Name changed by Gus Wiseman, Nov 15 2018

a(20) onwards from Ludovic Schwob, Oct 13 2023