A321652 Number of nonnegative integer matrices with sum of entries equal to n and no zero rows or columns, with weakly decreasing row sums and column sums.

1, 1, 5, 19, 107, 573, 4050, 29093, 249301, 2271020, 23378901, 257871081, 3132494380, 40693204728, 572089068459, 8566311524788, 137165829681775, 2327192535461323, 41865158805428687, 793982154675640340, 15863206077534914434, 332606431999260837036, 7309310804287502958322, 167896287022455809865568

Offset

0,3

Links

Ludovic Schwob, Table of n, a(n) for n = 0..39

Formula

Sum of coefficients in the expansions of all homogeneous symmetric functions in terms of monomial symmetric functions. In other words, if Sum_{|y| = n} h(y) = Sum_{|y| = n} c_y * m(y), then a(n) = Sum_{|y| = n} c_y.

Example

The a(3) = 19 matrices:
  [3] [2 1] [1 1 1]
.
  [2] [2 0] [1 1] [1 1 0] [1 0 1] [0 1 1]
  [1] [0 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]

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/@#]==Range[Max@@Last/@#], OrderedQ[Total/@prs2mat[#]], OrderedQ[Total/@Transpose[prs2mat[#]]]]&]], {n, 6}]

Crossrefs

Cf. A000219, A001970, A007716, A068313, A114736, A120733, A319646, A321645, A321653, A321654, A321655.

Sequence in context: A071828 A280067 A158615 * A088180 A370459 A206709

Adjacent sequences: A321649 A321650 A321651 * A321653 A321654 A321655

Keyword

nonn

Author

Gus Wiseman, Nov 15 2018

Extensions

a(10) onwards from Ludovic Schwob, Aug 29 2023

Status

approved