OFFSET
3,11
COMMENTS
For a k-gon to be nondegenerate, the longest side must be shorter than the sum of the remaining sides.
T(n,k) = number of partitions of n into k parts (k >= 3) in which all parts are less than n/2. Also T(n,k) = number of partitions of 2*n into k parts in which all parts are even and less than n. - L. Edson Jeffery, Mar 19 2012
LINKS
T. D. Noe, Rows n=3..102 of triangle, flattened
G. E. Andrews, P. Paule and A. Riese, MacMahon's Partition Analysis IX: k-Gon Partitions, Bull. Austral. Math. Soc. 64 (2001), 321-329.
James East, Ron Niles, Integer polygons of given perimeter, arXiv:1710.11245 [math.CO], 2017.
FORMULA
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)).
EXAMPLE
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
MAPLE
b:= proc(n, i) option remember; `if`(n=0, [1], `if`(i<1, [],
zip((x, y)-> x+y, b(n, i-1), `if`(i>n, [],
[0, b(n-i, i)[]]), 0)))
end:
T:= n-> b(n, ceil(n/2)-1)[4..n+1][]:
seq(T(n), n=3..20); # Alois P. Heinz, Jul 15 2013
MATHEMATICA
Flatten[Table[p=IntegerPartitions[n]; Length[Select[p, Length[ # ]==k && #[[1]] < Total[Rest[ # ]]&]], {n, 3, 30}, {k, 3, n}]]
(* second program: *)
QP = QPochhammer; T[n_, k_] := SeriesCoefficient[x^k*(1/QP[x, x, k] + x^(k - 2)/((x-1)*QP[x^2, x^2, k-1])), {x, 0, n}]; Table[T[n, k], {n, 3, 16}, {k, 3, n}] // Flatten (* Jean-François Alcover, Jan 08 2016 *)
CROSSREFS
KEYWORD
AUTHOR
T. D. Noe, Oct 24 2006
STATUS
approved