cp's OEIS Frontend

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.

%I A320212 #18 Oct 08 2018 08:07:54
%S A320212 1,2,12,256,26888,148958
%N 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.
%C A320212 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).
%C A320212 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).
%H A320212 József Balogh, Béla Bollobás and Robert Morris, <a href="https://projecteuclid.org/euclid.aop/1248182140">Bootstrap percolation in three dimensions</a>, Ann. Probab. 37 (2009), no. 4, 1329-1380.
%H A320212 L. Shapiro and A. B. Stephens, <a href="https://doi.org/10.1137/0404025">Bootstrap percolation, the Schröder numbers and the N-kings problem</a>, SIAM J. Discrete Math., Vol. 4 (1991), pp. 275-280.
%e A320212 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.
%Y A320212 Cf. A006318, A146971.
%K A320212 nonn,more
%O A320212 1,2
%A A320212 _Jonathan Noel_, Oct 07 2018