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.

1, 0, 2, 0, 4, 6, 0, 1, 45, 24, 0, 0, 90, 432, 120, 0, 0, 78, 2248, 4200, 720, 0, 0, 36, 5776, 43000, 43200, 5040, 0, 0, 9, 9066, 222925, 755100, 476280, 40320, 0, 0, 1, 9696, 727375, 6700500, 13003620, 5644800, 362880, 0, 0, 0, 7480, 1674840

Offset: 1

Author

Ralf Stephan, Mar 27 2005

Keywords

• nonn
• tabl

Examples

			1
0,2
0,4,6
0,1,45,24
0,0,90,432,120
0,0,78,2248,4200,720
0,0,36,5776,43000,43200,5040
0,0,9,9066,222925,755100,476280,40320
0,0,1,9696,727375,6700500,13003620,5644800,362880
0,0,0,7480,1674840,37638036,179494350,226262400,71850240,3628800
		

Links

• M. Maia and M. Mendez, On the arithmetic product of combinatorial species, arXiv:math/0503436 [math.CO], 2005.

Crossrefs

Right-edge diagonals include A000142, A055602, A055603. Row sums are in A104602.

Column sums are in A048291. The triangle read by columns = A055599.

Cf. A049311, A054976, A057150, A057151, A101370, A120732, A120733, A138178, A316983, A319616.

Programs

Mathematica

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 *)
Table[Length[Select[Subsets[Tuples[Range[n],2],{n}],Union[First/@#]==Union[Last/@#]==Range[k]&]],{n,6},{k,n}] (* Gus Wiseman, Nov 14 2018 *)

Formula

T(r, n) = Sum{l>=r, Sum{d|l, (-1)^(2n-d-l/d)*C(n, d)*C(n, l/d)*C(l, r) }}.

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