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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. 6
1, 1, 5, 19, 107, 573, 4050, 29093, 249301, 2271020 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

LINKS

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

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 A206709 A199480

Adjacent sequences: A321649 A321650 A321651 * A321653 A321654 A321655

KEYWORD

nonn,more

AUTHOR

Gus Wiseman, Nov 15 2018

STATUS

approved

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Last modified January 29 13:51 EST 2023. Contains 359923 sequences. (Running on oeis4.)