# [seqfan] Symmetrical Hamiltonian cycles on 2n*2n square grids

ed.wynn ed.wynn at zoho.com
Wed Feb 5 22:07:19 CET 2014

```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 http://list.seqfan.eu/pipermail/seqfan/2012-March/009134.html.)

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

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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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