A104601 - OEIS

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

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OFFSET

1,3

LINKS

Table of n, a(n) for n=1..50.

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

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

EXAMPLE

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

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 *)

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.

Sequence in context: A265820 A096984 A213723 * A233673 A319931 A192133

Adjacent sequences: A104598 A104599 A104600 * A104602 A104603 A104604

KEYWORD

nonn,tabl

AUTHOR

Ralf Stephan, Mar 27 2005

STATUS

approved