A230879 -id:A230879 - cp's OEIS frontend

A230878 Irregular triangle read by rows: T(n,k) = number of 2-packed n X n matrices with exactly k nonzero entries (0 <= k <= n^2).

1, 0, 2, 0, 0, 8, 32, 16, 0, 0, 0, 48, 720, 2880, 4992, 4608, 2304, 512, 0, 0, 0, 0, 384, 13824, 143872, 739328, 2320896, 4964352, 7659520, 8749056, 7421952, 4587520, 1966080, 524288, 65536, 0, 0, 0, 0, 0, 3840, 268800, 5504000, 57068800, 372416000

Offset: 0

N. J. A. Sloane, Nov 09 2013

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.

			Triangle begins:
1
0 2
0 0 8 32 16
0 0 0 48 720 2880 4992 4608 2304 512
...

Andrew Howroyd, Table of n, a(n) for n = 0..662 (rows 0..12 flattened)
H. Cheballah, S. Giraudo, R. Maurice, Combinatorial Hopf algebra structure on packed square matrices, arXiv preprint arXiv:1306.6605 [math.CO], 2013.

Row sums are A230879.
Column sums are A230880.
Cf. A000165, A055599.

p[k_, n_, l_] := Sum[(-1)^(i+j)*Binomial[n, i]*Binomial[n,j]*Binomial[i*j, l]*k^l, {i, 0, n}, {j, 0, n}];
T[n_, k_] := p[2, n, k];
Table[T[n, k], {n, 0, 5}, {k, 0, n^2}] // Flatten (* Jean-François Alcover, Oct 08 2017, translated from PARI *)

\\ T(n,k) = p(2,n,k) (see Cheballah et al. ref).
p(k,n,l) = {sum(i=0, n, sum(j=0, n, (-1)^(i+j) * binomial(n,i) * binomial(n,j) * binomial(i*j,l) * k^l))}
for (n=0,5, for(k=0,n^2, print1(p(2,n,k), ", ")); print); \\ Andrew Howroyd, Sep 20 2017

From Andrew Howroyd, Sep 20 2017: (Start)
T(n, k) = Sum_{i=0..n} Sum_{j=0..n} (-1)^(i+j) * binomial(n,i) * binomial(n,j) * binomial(i*j,k) * 2^k.
T(n, k) = 0 for n > k.
T(n, n) = A000165(n).
(End)

Terms a(18) and beyond from Andrew Howroyd, Sep 20 2017

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

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane, Nov 09 2013

nonn

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.

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.

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

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 *)

\\ 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

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)

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

cp's OEIS Frontend

A230878 Irregular triangle read by rows: T(n,k) = number of 2-packed n X n matrices with exactly k nonzero entries (0 <= k <= n^2).

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