[seqfan] Re: Two-fold symmetry sequence
Frederick Schneider
frederick.william.schneider at gmail.com
Tue Dec 30 04:14:31 CET 2008
Hello Joseph,
Thanks for your insight and pointers. I'll do some digging now.
I didn't see the image in your last mail. If you could attach that or
provide a link, that'd be great.
Best,
Fred
2008/12/29 Joseph S. Myers <jsm at polyomino.org.uk>
> On Mon, 29 Dec 2008, Frederick Schneider wrote:
>
> > Here's an image explaining what I mean:
> >
> > http://www.bosola.org/grandpa/imgs/2-foldSymmetry.jpg
> >
> > Has anyone seen a similar sequence? If not, I'll try to extend this at
> > least a few terms out.
>
> This (free polyominoes with at least c2 symmetry) would be the sum of the
> columns "all", "axis 2", "rotate 2", "diag 2" and "rotate" from
> Redelmeier's Table 3 (in Counting polyominoes: yet another attack,
> Discrete Math. 36 (1981) 191-203) which respectively count those whose
> symmetries are exactly d4, d2 (horizontal and vertical reflections), c4,
> d2 (diagonal reflections) and c2. "axis 2" is A056877, "diag 2" is
> A056878, "rotate" is A006747. The columns for polyominoes without c2
> symmetry are also present , "axis" (A006746), "diag" (A006748), "none"
> (A006749). For some reason, "all" and "rotate 2" seem to be missing.
> (Those sequences are zero for n = 2 or 3 mod 4 so it's not clear what
> versions should be present - for all n? n = 0 and 1 mod 4 only? n = 0
> mod 4 and n = 1 mod 4 in separate sequences? But none of these variants
> seem to be present.) An image of the table is attached.
>
> Tomas Oliveira e Silva could likely provide versions of the missing
> sequences up to n=28 (Redelmeier gives them to n=25), though I can't find
> counts of symmetric polyominoes on his webpages. The figures for "at
> least" a given symmetry, such as your sequence, would seem to be worth
> adding as well if not already present.
>
> Incidentally, it should be possible to implement Redelmeier's algorithms
> for counting symmetric polyominoes and to combine those with the
> enumeration of fixed polyominoes (A001168) up to n=56 by the transfer
> matrix method in order to extend the numbers for free polyominoes
> (A000105) up to n=56 or pretty near that (from the present limit of n=28);
> the counts involving symmetry groups of order 4 or 8 should be extendable
> well beyond that. Though counting symmetric polyominoes is more
> complicated than counting free polyominoes, and Redelmeier doesn't give
> the details of the algorithms he used for counting rings where
> connectivity issues come in. (Converting numbers of fixed polyominoes
> with "at least" a given symmetry, as given by these algorithms, to numbers
> of fixed polyominoes with exactly a given symmetry, and so to free
> polyominoes with exactly a given symmetry, is routine linear manipulation
> of obvious formulae giving "at least" in terms of "exact".)
>
> --
> Joseph S. Myers
> jsm at polyomino.org.uk
>
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