[seqfan] Re: A045310 as partitions

[Neil Sloane]

Hugo, Maybe you could add a comment or two
to A045310 saying something like: Equivalently, this is the number of
decompositions of an n-dimensional cube of size 2 into unit cubes
(1X1X...X1) and "dominoes" (2X1X1X...X1).", if that is what you are saying
(I'm not sure what a polycube is either)!

And by all means add your two programs.

Best regards
Neil

Neil J. A. Sloane, President, OEIS Foundation.
11 South Adelaide Avenue, Highland Park, NJ 08904, USA.
Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ.
Phone: 732 828 6098; home page: http://NeilSloane.com
Email: njasloane at gmail.com


On Mon, Nov 28, 2016 at 5:43 AM, <hv at crypt.org> wrote:

> 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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