[seqfan] Re: Need help naming another "regular n-gon with all diagonals" sequence
Arthur O'Dwyer
arthur.j.odwyer at gmail.com
Mon Sep 25 16:45:42 CEST 2023
https://oeis.org/A007678 "Number of regions in *[sic]* regular n-gon with
all diagonals drawn"
https://oeis.org/A187781 "Number of noncongruent polygonal regions in a
regular n-gon with all diagonals drawn"
https://oeis.org/A363979 "Number of nonsimilar polygonal regions in a
regular n-gon with all diagonals drawn"
I suggest that all of these three and your extensions fit the pattern
"Count of regions in a regular n-gon with all its diagonals, up to
*[symmetry]*"
where *[symmetry]* is "congruence," "similarity," "the dihedral
symmetries," etc. etc.
(Maybe "up to" is too vague and/or off-target; if so, I hope a
mathematician will correct me. I'm pretty sure *some* phrase means what we
mean here.)
And personally I like your shorter description, even if it is a little
oblique:
> The maximum number of colors with which the regions can be colored while
retaining all of the dihedral symmetries of the n-gon.
similarly,
- A187781 The maximum number of colors with which the regions can be
colored such that any two congruent regions have the same color.
- A363979 The maximum number of colors with which the regions can be
colored such that any two similar regions have the same color.
- Axyzxyz The maximum number of colors with which the regions can be
colored while retaining all of the dihedral symmetries of the n-gon.
The only potential confusion here is that usually "coloring" means no two
*adjacent* regions can ever have the same color; whereas a(4) for all of
these sequences is 1 because we intend to color all four regions the same
color even though they border each other. Perhaps saying "painted" instead
of "colored" would avoid this potential confusion, or perhaps it would just
add more confusion about why on earth we're saying "painted." ;)
Cheers,
Arthur
On Sun, Sep 24, 2023 at 11:11 PM Chris Scussel via SeqFan <
seqfan at list.seqfan.eu> wrote:
> Greetings Seqfans,
>
> Recently I have corrected and extended A187781 (counting noncongruent
> regions)
> and created A363979 (counting nonsimilar regions), both for "regular
> n-gons with all
> diagonals". I am working on a related new sequence (currently unnumbered)
> that
> I am having trouble describing succinctly.
>
> The concept is quite simple: the number of families of congruent
> regions,
> where congruence follows directly from dihedral symmetries of the n-gon.
> This sequence
> is the same as A187781 until n=12, at which point a "sporadic" congruence
> (i.e., one not caused by dihedral symmetries) occurs. For n>12 such
> congruences
> become more frequent and numerous.
>
> Among the potential descriptions I've considered are the following:
>
>
> A wordy description:
>
> The number of equivalence classes in an equivalence relation on
> the regions,
> where two regions are equivalent iff they map into each other via
> dihedral
> symmetries of the n-gon
>
> One less wordy but also less to the point (and possibly not even correct):
>
> The maximum number of colors with which the regions can be colored
> while
> retaining all of the dihedral symmetries of the n-gon.
>
> Neither of these include the necessary phrase "regular n-gon with all
> diagonals",
> which would make them even longer.
>
> I would appreciate any suggestions.
>
>
> Thanks and Regards,
>
> Chris
>
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>
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