A321515 Number of nonnegative integer matrices with sum of entries equal to n, no zero rows or columns, and distinct rows and columns.

1, 1, 3, 19, 137, 1209, 12899, 160395, 2276229, 36323217, 643848837, 12551081501, 266868756473, 6146455542737, 152439235077709, 4050427673024753, 114791270281213209, 3456412742412516649, 110191808168628510207, 3708004806262196242699, 131339701217968663631857

Offset

0,3

Links

Andrew Howroyd, Table of n, a(n) for n = 0..40

Example

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

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[#], UnsameQ@@Transpose[prs2mat[#]]]&]], {n, 5}]

Prog

(PARI) \\ Q(m, n, wf) defined in A321588.
seq(n)={my(R=vectorv(n, m, Q(m, n, w->1/(1 - y^w) + O(y*y^n)))); for(i=2, #R, R[i] -= i*R[i-1]); Vec(1 + vecsum(vecsum(R)))} \\ Andrew Howroyd, Jan 24 2024

Crossrefs

Cf. A007716, A120733, A283877, A316980, A319559, A321446, A321586, A321588.

Sequence in context: A278189 A221297 A138513 * A094661 A094662 A321349

Adjacent sequences: A321512 A321513 A321514 * A321516 A321517 A321518

Keyword

nonn

Author

Gus Wiseman, Nov 13 2018

Extensions

a(7) onwards from Andrew Howroyd, Jan 20 2024

Status

approved