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A104601 Triangle T(r,n) read by rows: number of n X n (0,1)-matrices with exactly r entries equal to 1 and no zero row or columns.

%I A104601 #16 Nov 15 2018 08:39:43
%S A104601 1,0,2,0,4,6,0,1,45,24,0,0,90,432,120,0,0,78,2248,4200,720,0,0,36,
%T A104601 5776,43000,43200,5040,0,0,9,9066,222925,755100,476280,40320,0,0,1,
%U A104601 9696,727375,6700500,13003620,5644800,362880,0,0,0,7480,1674840
%N A104601 Triangle T(r,n) read by rows: number of n X n (0,1)-matrices with exactly r entries equal to 1 and no zero row or columns.
%H A104601 M. Maia and M. Mendez, <a href="https://arxiv.org/abs/math/0503436">On the arithmetic product of combinatorial species</a>, arXiv:math/0503436 [math.CO], 2005.
%F A104601 T(r, n) = Sum{l>=r, Sum{d|l, (-1)^(2n-d-l/d)*C(n, d)*C(n, l/d)*C(l, r) }}.
%F A104601 E.g.f.: Sum(((1+x)^n-1)^n*exp((1-(1+x)^n)*y)*y^n/n!,n=0..infinity). - _Vladeta Jovovic_, Feb 24 2008
%e A104601 1
%e A104601 0,2
%e A104601 0,4,6
%e A104601 0,1,45,24
%e A104601 0,0,90,432,120
%e A104601 0,0,78,2248,4200,720
%e A104601 0,0,36,5776,43000,43200,5040
%e A104601 0,0,9,9066,222925,755100,476280,40320
%e A104601 0,0,1,9696,727375,6700500,13003620,5644800,362880
%e A104601 0,0,0,7480,1674840,37638036,179494350,226262400,71850240,3628800
%t A104601 t[r_, n_] := Sum[ Sum[ (-1)^(2n - d - k/d)*Binomial[n, d]*Binomial[n, k/d]*Binomial[k, r], {d, Divisors[k]}], {k, r, n^2}]; Flatten[ Table[t[r, n], {r, 1, 10}, {n, 1, r}]] (* _Jean-François Alcover_, Jun 27 2012, from formula *)
%t A104601 Table[Length[Select[Subsets[Tuples[Range[n],2],{n}],Union[First/@#]==Union[Last/@#]==Range[k]&]],{n,6},{k,n}] (* _Gus Wiseman_, Nov 14 2018 *)
%Y A104601 Right-edge diagonals include A000142, A055602, A055603. Row sums are in A104602.
%Y A104601 Column sums are in A048291. The triangle read by columns = A055599.
%Y A104601 Cf. A049311, A054976, A057150, A057151, A101370, A120732, A120733, A138178, A316983, A319616.
%K A104601 nonn,tabl
%O A104601 1,3
%A A104601 _Ralf Stephan_, Mar 27 2005