A230880 Number of 2-packed matrices with exactly n nonzero entries.

1, 2, 8, 80, 1120, 20544, 463744, 12422656, 384947200, 13541822464, 533049493504, 23210958688256, 1107652218822656, 57482801016422400, 3223015475535380480, 194157345516262588416, 12505948470244176953344, 857670052436844788318208, 62395270194815987194789888

Offset: 0

Author

N. J. A. Sloane, Nov 09 2013

Keywords

• nonn

Comments

A k-packed matrix of size n X n is a matrix with entries in the alphabet A_k = {0,1, ..., k} such that each row and each column contains at least one nonzero entry.

Links

• Andrew Howroyd, Table of n, a(n) for n = 0..100
• H. Cheballah, S. Giraudo, R. Maurice, Combinatorial Hopf algebra structure on packed square matrices, arXiv preprint arXiv:1306.6605 [math.CO], 2013.

Crossrefs

Cf. A048291, A055599, A230878, A230879, A104602.

Programs

Mathematica

b[n_] := Sum[StirlingS1[n, k]*Sum[(m!)^2*StirlingS2[k, m]^2, {m, 0, k}], {k, 0, n}]/n!;
a[n_] := 2^n*b[n];
Table[a[n], {n, 0, 18}] (* Jean-François Alcover, Oct 08 2017, translated from PARI *)

PARI

\\ here b(n) is A104602.
b(n) = {sum(m=0, n, sum(k=0, n, stirling(n,k,1) * m!^2 * stirling(k,m,2)^2)) / n!}
a(n) = 2^n * b(n); \\ Andrew Howroyd, Sep 20 2017

Formula

Cheballah et al. give an explicit formula.

From Andrew Howroyd, Sep 20 2017: (Start)

a(n) = Sum_{r=1..n} Sum_{i=0..r} Sum_{j=0..r} (-1)^(i+j) * binomial(r,i) * binomial(r,j) * binomial(i*j,n) * 2^n.

a(n) = 2^n * A104602(n).

(End)

Extensions

Terms a(9) and beyond from Andrew Howroyd, Sep 20 2017