A321586 Number of nonnegative integer matrices with sum of entries equal to n, no zero rows or columns, and distinct rows (or distinct columns).

1, 1, 4, 26, 204, 1992, 23336, 318080, 4948552, 86550424, 1681106080, 35904872576, 836339613984, 21100105791936, 573194015723840, 16681174764033728, 517768654898701120, 17074080118403865856, 596117945858272441408, 21967609729338776864384, 852095613819396775627200

Offset

0,3

Links

Alois P. Heinz, Table of n, a(n) for n = 0..200

Example

The a(3) = 26 matrices:
  [3][21][12][111]
.
  [2][20][11][11][110][101][1][10][10][100][02][011][01][01][010][001]
  [1][01][10][01][001][010][2][11][02][011][10][100][20][11][101][110]
.
  [100][100][010][010][001][001]
  [010][001][100][001][100][010]
  [001][010][001][100][010][100]

Maple

C:= binomial:
b:= proc(n, i, k, p) option remember; `if`(n=0, p!, `if`(i<1, 0, add(
      b(n-i*j, min(n-i*j, i-1), k, p+j)*C(C(k+i-1, i), j), j=0..n/i)))
    end:
a:= n-> add(add(b(n$2, i, 0)*(-1)^(k-i)*C(k, i), i=0..k), k=0..n):
seq(a(n), n=0..21);  # Alois P. Heinz, Sep 16 2019

Mathematica

multsubs[set_, k_]:=If[k==0, {{}}, Join@@Table[Prepend[#, set[[i]]]&/@multsubs[Drop[set, i-1], k-1], {i, Length[set]}]];
prs2mat[prs_]:=Table[Count[prs, {i, j}], {i, Union[First/@prs]}, {j, Union[Last/@prs]}];
Table[Length[Select[multsubs[Tuples[Range[n], 2], n], And[Union[First/@#]==Range[Max@@First/@#], Union[Last/@#]==Range[Max@@Last/@#], UnsameQ@@prs2mat[#]]&]], {n, 5}]

Crossrefs

Cf. A007716, A049311, A101370, A120733, A283877, A316980, A321446, A321587.

Row sums of A327245.

Sequence in context: A118971 A369102 A124554 * A206391 A368968 A369511

Adjacent sequences: A321583 A321584 A321585 * A321587 A321588 A321589

Keyword

nonn

Author

Gus Wiseman, Nov 13 2018

Extensions

a(7)-a(20) from Alois P. Heinz, Sep 16 2019

Status

approved