[seqfan] A045310 as partitions

hv at crypt.org hv at crypt.org
Mon Nov 28 11:43:02 CET 2016


I've recently been considering partitions of n-cubes into n-dimensional
analogues of polyominoes, and stumbled on A045310 as the number of
partitions of an n-cube of side 2 into polyominoes of size at most 2.

The identity became obvious once I understood what the Hosoya index
represented, but I feel it's worth a comment; however I don't know
the terminology for n-dimensional polyominoes, so I'm not sure how
to phrase such a comment - Wikipedia has brief mentions of "polycubes"
and "polyhypercubes", while Mathworld confusingly defines polycube as
"Three-dimensional generalization of the polyominoes to n dimensions".

The code I wrote may also be useful - it verifies the existing values,
though it would struggle to extend to n=7. Proof of concept is in Perl,
and a version fast enough to be useful is in C:
  https://github.com/hvds/seq/blob/master/part/find2
  https://github.com/hvds/seq/blob/master/part/find2c.c

Hugo


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