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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

Jonathan Noel, Oct 07 2018

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).

			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.

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.

Cf. A006318, A146971.

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User: Jonathan Noel

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.

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