Re^2: Counting n-gons

[Richard Mathar]

tdn> From seqfan-owner at ext.jussieu.fr  Mon Oct 23 17:27:12 2006
tdn> Return-Path: <seqfan-owner at ext.jussieu.fr>
tdn> Date: Mon, 23 Oct 2006 08:26:28 -0700
tdn> To: "David Wilson" <davidwwilson at comcast.net>,
tdn>         "Sequence Fans" <seqfan at ext.jussieu.fr>
tdn> From: "T. D. Noe" <noe at sspectra.com>
tdn> Subject: Re: Counting n-gons
tdn> 
tdn> At 10:31 AM -0400 10/23/06, David Wilson wrote:
tdn> >Let f(n,p) be the number of non-congruent integer-sided n-gons with
tdn> >perimeter p. Then f(3,p) = A005044.
tdn> >
tdn> >Can we come up with a general formula/recurrence for f(n, p)?
tdn> 
tdn> For 4-gons, see A062890.
tdn> For 5-gons, see A069906.
tdn> For 6-gons, see A069907.
tdn> 
tdn> Tony

This is not clear to me: if we form partitions of n into as many
parts as the n-gon is having sides, we still can sometimes generate non-congruent
polygons with the same partitioning by flipping two sides at the time
inside-out (changing convex to concave or vice versa in this region).
Therefore A069906 and A069907 may not (and do not claim to) refer
to non-congruent counts.
(I admit that I did not read the Andrews et al reference yet...)

Recurrences for increase of the side number, n ->n+1 would be difficult
to find: imagine a 4-gon is split into two 3-gons by connecting two
opposite corners with a straight line; the two pieces are 3-gons but
this diagonal of the 4-gon would generally not have an integer-sized
diagonal to create 3-gons that match the definitions..

Richard