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

Author

Gus Wiseman, Nov 13 2018

Keywords

• nonn

Examples

			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]
		

Links

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

Crossrefs

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

Programs

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}]

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

Extensions

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