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

%I A321515 #34 Jan 24 2024 18:32:10
%S A321515 1,1,3,19,137,1209,12899,160395,2276229,36323217,643848837,
%T A321515 12551081501,266868756473,6146455542737,152439235077709,
%U A321515 4050427673024753,114791270281213209,3456412742412516649,110191808168628510207,3708004806262196242699,131339701217968663631857
%N A321515 Number of nonnegative integer matrices with sum of entries equal to n, no zero rows or columns, and distinct rows and columns.
%H A321515 Andrew Howroyd, <a href="/A321515/b321515.txt">Table of n, a(n) for n = 0..40</a>
%e A321515 The a(3) = 19 matrices:
%e A321515   [3] [2 1] [1 2]
%e A321515 .
%e A321515   [2] [2 0] [1 1] [1 1] [1] [1 0] [1 0] [0 2] [0 1] [0 1]
%e A321515   [1] [0 1] [1 0] [0 1] [2] [1 1] [0 2] [1 0] [2 0] [1 1]
%e A321515 .
%e A321515   [1 0 0] [1 0 0] [0 1 0] [0 1 0] [0 0 1] [0 0 1]
%e A321515   [0 1 0] [0 0 1] [1 0 0] [0 0 1] [1 0 0] [0 1 0]
%e A321515   [0 0 1] [0 1 0] [0 0 1] [1 0 0] [0 1 0] [1 0 0]
%t A321515 multsubs[set_,k_]:=If[k==0,{{}},Join@@Table[Prepend[#,set[[i]]]&/@multsubs[Drop[set,i-1],k-1],{i,Length[set]}]];
%t A321515 prs2mat[prs_]:=Table[Count[prs,{i,j}],{i,Union[First/@prs]},{j,Union[Last/@prs]}];
%t A321515 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}]
%o A321515 (PARI) \\ Q(m,n,wf) defined in A321588.
%o A321515 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
%Y A321515 Cf. A007716, A120733, A283877, A316980, A319559, A321446, A321586, A321588.
%K A321515 nonn
%O A321515 0,3
%A A321515 _Gus Wiseman_, Nov 13 2018
%E A321515 a(7) onwards from _Andrew Howroyd_, Jan 20 2024