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

1,2

Comments

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

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

Example

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)

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 *)

Crossrefs

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

Sequence in context: A073479 A147690 A350830 * A174661 A207161 A203121

Adjacent sequences: A068310 A068311 A068312 * A068314 A068315 A068316

Keyword

nonn

Author

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

Extensions

Name changed by Gus Wiseman, Nov 15 2018

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

Status

approved