[seqfan] Re: Symmetrical Hamiltonian cycles on 2n*2n square grids
njasloane at gmail.com
Thu Feb 6 13:33:34 CET 2014
Ed, I think the answer is a definite Yes.
In other counting problems that are treated in the OEIS
the subsidiary sequences (classifying the results
according to symmetry group) have turned out
to be useful later.
So please go ahead and submit them.
On Wed, Feb 5, 2014 at 4:07 PM, ed.wynn <ed.wynn at zoho.com> wrote:
> Hi Seqfans,
> I have recently added to sequences https://oeis.org/A227257 and
> https://oeis.org/A227005, which are counts of Hamiltonian cycles on 2n*2n
> grids. Specifically, these have 4 and 2 orbits under the symmetry group of
> the square. (These are also related to https://oeis.org/A209077 and an
> oldish thread
> My question is whether it is worthwhile to send in new sequences
> subdividing these two sequences into specific symmetries. Cycles (properly
> speaking, isomorphism classes of cycles) counted in A227257 have either
> 180-degree rotational symmetry or a single axis of reflective symmetry (and
> no others); those in A227005 have either 90-degree rotational symmetry or
> two axes of reflective symmetry (which inevitably bring 180-degree
> rotational symmetry as well).
> If people are interested in the sequences, they can find them in the Arxiv
> paper that I've referenced in the entries: http://arxiv.org/abs/1402.0545.
> Also, for completeness, I'll put them at the end of this email.
> I would also like to ask the analogous question for
> https://oeis.org/A224239 (Number of inequivalent ways to cut an n X n
> square into squares with integer sides). This is also divided into
> examples with specified orbits under symmetry: A226978(n) + A226979(n) +
> A226980(n) + A226981(n) = A224239(n). Should these be subdivided into
> specified symmetries?
> Thanks for your attention. Best regards,
> Ed Wynn
> Here are the counts of isomorphism classes with specified symmetries (and
> no others), with offset 1:
> Subdivision of A227257:
> 180-degree rotation: 0, 0, 5, 366, 129871, 174041330, 1343294003351,
> 41725919954578785, 7159149948562719664049, 5065741493544986113047994120.
> one reflection: 0 , 1, 19, 1394, 281990, 377205809, 1539951848735,
> 44222409563201991, 3842818845468254120853, 2396657968905952750257244144.
> Subdivision of A227005:
> 90-degree rotation: 0, 0, 1, 0, 102, 0, 255359, 0, 15504309761, 0.
> two reflections and 180-degree rotation: 0, 1, 3, 20, 244, 6891, 378813,
> 47917598, 12118420172, 6998287399637.
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Neil J. A. Sloane, President, OEIS Foundation
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