[seqfan] Re: Symmetrical Hamiltonian cycles on 2n*2n square grids
njasloane at gmail.com
Thu Feb 6 13:39:43 CET 2014
Ed, there is always a question which comes up at this point. Some
of the subgroup sequences will have alternate zeros, as in this one
from your email:
90-degree rotation: 0, 0, 1, 0, 102, 0, 255359, 0, 15504309761, 0.
The question is, should one skip the zeros, which is the usual convention
in the OEIS, or should one include them,
so that in the end one can say that the main sequence is a simple
sum a(n) = b(n) + c(n) + d(n) + ...
rather than having to say
a(n) = b(n) + c(2n) + d(n) + ...
On the whole I think it is better to stay with
the usual OEIS convention, and leave out the
alternating 0's. So 0,1,102, ... in
On Thu, Feb 6, 2014 at 7:33 AM, Neil Sloane <njasloane at gmail.com> wrote:
> 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.
> Best regards
> 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/A209077and 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.
>> Seqfan Mailing list - http://list.seqfan.eu/
> Dear Friends, I have now retired from AT&T. New coordinates:
> 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
Dear Friends, I have now retired from AT&T. New coordinates:
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
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