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A320212
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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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0
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OFFSET
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1,2
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COMMENTS
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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).
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LINKS
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EXAMPLE
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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.
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CROSSREFS
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KEYWORD
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nonn,more
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AUTHOR
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STATUS
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approved
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