A230879 Number of 2-packed n X n matrices.

1, 2, 56, 16064, 39156608, 813732073472, 147662286695991296, 237776857718965784182784, 3425329990022686416530808209408, 443021337239562368918979788606843912192, 515203019085226443540506018909263027730003787776

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..30
• H. Cheballah, S. Giraudo, R. Maurice, Combinatorial Hopf algebra structure on packed square matrices, arXiv preprint arXiv:1306.6605 [math.CO], 2013.

Crossrefs

Row sums of A230878.

Cf. A048291, A055599, A230880, A104602.

Programs

Mathematica

p[k_, n_] := Sum[(-1)^(i + j)*Binomial[n, i]*Binomial[n, j]*(k + 1)^(i*j), {i, 0, n}, {j, 0, n}];
a[n_] := p[2, n];
Table[a[n], {n, 0, 10}] (* Jean-François Alcover, Oct 08 2017, translated from PARI *)

PARI

\\ here p(k,n) is number of k-packed matrices of size n.
p(k,n)={sum(i=0, n, sum(j=0, n, (-1)^(i+j) * binomial(n,i) * binomial(n,j) * (k+1)^(i*j)))}
a(n) = p(2,n); \\ Andrew Howroyd, Sep 20 2017

Formula

Cheballah et al. give an explicit formula.

a(n) = Sum_{i=0..n} Sum_{j=0..n} (-1)^(i+j) * binomial(n,i) * binomial(n,j) * 3^(i*j). - Andrew Howroyd, Sep 20 2017

Extensions

Terms a(7) and beyond from Andrew Howroyd, Sep 20 2017