Counting n-gons

[T. D. Noe]

At 5:13 PM -0400 10/23/06, franktaw at netscape.net wrote:
>There's a problem with this formulation.  Triangles are determined up
>to congruence by their side lenths, but polygons with more sides are
>not.  (Consider rhombus vs. square.)
>
>To make it work, you have to define it purely in terms of edge lengths:
>sequences of n positive integers, totalling p, whose largest value is
>less than the sum of the others (equivalently, less than p/2); up to
>equivalence under rotation and reflection.
>
>Franklin T. Adams-Watters
>
>
>-----Original Message-----
>From: davidwwilson at comcast.net
>
>Let f(n,p) be the number of non-congruent integer-sided n-gons with
>perimeter p. Then f(3,p) = A005044.
> 
>Can we come up with a general formula/recurrence for f(n, p)?


I just submitted A124278, triangle of the number of k-gons having perimeter
n.  Included is a generating function for each column of the triangle.
There is also a b-file for rows n=3..102 of the triangle.  Here is the
internal format:

%S A124278 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 1, 1, 1, 3, 2, 2, 1, 1,
3, 4, 4, 3, 2, 1, 1, 2, 5, 5, 4, 3, 2, 1, 1, 4, 7, 8, 6, 5, 3, 2, 1, 1, 3,
8, 9, 9, 6, 5, 3, 2, 1, 1, 5, 11, 14, 12, 10, 7, 5, 3, 2, 1, 1, 4, 12, 16,
16, 13, 10, 7, 5, 3, 2, 1, 1, 7, 16, 23, 22, 19, 14, 11, 7, 5, 3, 2, 1, 1,
5, 18, 25, 28, 24, 20, 14, 11, 7, 5, 3, 2, 1, 1, 8, 23, 35, 37, 34, 27, 21,
15, 11, 7, 5, 3, 2, 1, 1
%N A124278 Triangle of the number of nondegenerate k-gons having perimeter
n, for k=3..n.
%C A124278 For a k-gon to be nondegenerate, the longest side must be
shorter than the sum of the remaining sides. Column k=3 is A005044, column
k=4 is A062890, column k=5 is A069906, and column k=6 is A069907.
%H A124278 G.E. Andrews, P. Paule, and A. Riese, <a
href="http://www.risc.uni-linz.ac.at/publications/download/risc_163/PAIX.pdf">MacMahon's
Partition Analysis IX: k-Gon Partitions, Bull. Austral. Math. Soc. 64
(2001), 321-329.</a>
%H A124278 T. D. Noe, <a
href="http://www.research.att.com/~njas/sequences/b124278.txt">Rows
n=3..102 of triangle, flattened</a>
%F A124278 G.f. for column k is x^k/(product_{i=1..k} 1-x^i) -
x^(2k-2)/(1-x)/(product_{i=1..k-1} 1-x^(2i))
%e A124278 For polygons having perimeter 7: 2 triangles, 2 quadrilaterals,
2 pentagons, 1 hexagon, and 1 heptagon.  The triangle begins
1
0 1
1 1 1
1 1 1 1
2 2 2 1 1
1 3 2 2 1 1
%t A124278 Needs["DiscreteMath`Combinatorica`"]; Table[p=Partitions[n];
Length[Select[p, Length[#]==k && #[[1]]<Total[Rest[#]]&]], {n,3,30},
{k,3,n}]
%O A124278 3

Tony