A320212 Number of binary n X n X n permutation arrays (all projections onto 2-dimensional faces yield the all-ones matrix) which yield the all-ones array when repeatedly changing a 0 with three 1 neighbors to 1.

1, 2, 12, 256, 26888, 148958

Offset

1,2

Comments

This can be phrased as the number of n X n X n permutation arrays which percolate with respect to the 3-neighbor bootstrap percolation rule in the n X n X n grid. C.f. Balogh, Bollobás and Morris (2009).

The analogous sequence for n X n permutation arrays with respect to 2-neighbor bootstrap percolation is enumerated by the Large Schröder numbers A006318. See Shapiro and Stephens (1991).

Links

Table of n, a(n) for n=1..6.

József Balogh, Béla Bollobás and Robert Morris, Bootstrap percolation in three dimensions, Ann. Probab. 37 (2009), no. 4, 1329-1380.

L. Shapiro and A. B. Stephens, Bootstrap percolation, the Schröder numbers and the N-kings problem, SIAM J. Discrete Math., Vol. 4 (1991), pp. 275-280.

Example

One example of such an array is the n X n X n array in which the (i,j,k) entry is 1 if i+j+k is 0 mod n. For n=2 and n=3, the arrays counted by a(n) are precisely the (n-1)!n! arrays that are obtained from this example by permuting rows and columns. For larger n, more complicated examples exist.

Crossrefs

Cf. A006318, A146971.

Sequence in context: A132481 A369680 A217652 * A361749 A012549 A009610

Adjacent sequences: A320209 A320210 A320211 * A320213 A320214 A320215

Keyword

nonn,more

Author

Jonathan Noel, Oct 07 2018

Status

approved