[seqfan] Re: A187781 - Number of noncongruent polygonal regions in a regular n-gon with all diagonals drawn

[Chris Scussel]

Neil told me that my original post was full of question marks.
Upon investigating I found that it was UTF-8 instead of text,
and my use of spaces led to the problem. I've posted a
repaired version below.

Neil also asked some questions. I have computed the sequences
for C and D up to n=60; they first differ at n=12. The dihedral
group I refer to in the definition of D is in fact the one of order
2n when considering an n-gon (as K_n).

Many thanks for the responses. Today was an unusually busy
day, and I apologize for my delayed response.

Chris

Repost of original post follows


I believe I have found that the description for this sequence actually describes
a different, but related, sequence. I am hoping to get a second opinion from SeqFans.

Define an equivalence relation D on the polygonal regions, where two regions are
enquivalent iff they map into each other under the dihedral symmetries of the 12-gon.
Define another equivalence relation C where two regions are equivalent iff they are
congruent. It appears to me that the terms of A187781 count the equivalence
classes of D, while the description specifies the count of equivalence classes of C.
Certainly all of the regions in any equivalence class of D are congruent. 
Is it possible that regions from different equivalence classes of D are congruent?

I believe so. This first occurs at a(12), where there are 22 equivalence classes for D,
but only 21 for C. I have proved (by brute force analytic geometry, with help from
WolframAlpha, then again later with plane geometry) the congruence of regions from
two of the equivalence classes of D. Here is a description of a do-it-yourself illustration:

Number the vertices of the 12-gon consecutively from 0 to 11,
then draw the following chords: 
0 5
1 4
1 7
2 6
3 10
The resulting figure has two small, congruent triangles. It's easy to see
that they are similar, but a bit harder to prove that they are congruent.
These triangles remain intact as regions in the full dissection. They are
at different distances from the center of the 12-gon, and thus are not
in the same equivalence class of D.

An analogous pair occurs for a(16). Thus, a(12) and a(16) are too large
by one if they are supposed to be counting the equivalence classes of C.
Going beyond the currently given terms for A187781, for a(18) there are
three such pairs, as well as a triple. Additional congruences occur for
all even n >14 up to at least n=60. They do not occur for any odd n<=60.

I discovered this while testing a tessellation-coloring program, which has
since morphed into program to count these classes. The program has computed
the equivalence class counts for D and C up to a(60). It also computed the
numbers of segments, vertices, regions, and polygons from triangles through
9-gons, each of which agree with the corresponding OEIS sequence. This and
a several other checks give me confidence that the results are correct.

It seems to me that OEIS needs an additional sequence, so that there is one for
D and one for C. As I understand it, in situations where the terms and description
conflict the terms are taken to define the sequence. If that is the case then A187781
would be for D, and the new sequence for C. And perhaps even a third for a relation S,
where two regions are equivalent iff they are similar.

I am hoping someone on this list can either point out where I have gone wrong
or agree that I'm making sense. I find it a bit hard to believe that this is new.


Thanks and Kind Regards,

Chris Scussel