Generalized triangulations and diagonal-free subsets of stack polyominoes

• Authors: Jakob Jonsson
• Venue: Journal of Combinatorial Theory Series A
• Year: 2005

For n≥3, let Ωn be the set of line segments between vertices in a convex n-gon. For j≥1, a j-crossing is a set of j distinet and mutually intersecting line segments from Ωn such that all 2j endpoints are distinct. For k≥1, let Δn,k be the simplicial complex of subsets of Ωn not containing any (k + 1)-crossing. For example, Δn,k has one maximal set for each triangulation of the n-gon, Dress, Koolen, and Moulton were able to prove that all maximal sets in Δn,k have the same number k(2n - 2k - 1) of line segments. We demonstrate that the number of such maximal sets in counted by a k × k determinant of Catalan numbers. By the work of Desainte-Catherine and Viennot, this determinant is known to count quite a few other objects including fans of non-crossing Dyck paths. We gerneralize our result to a larger class of simplicial complexes including some of the complexes appearing in the work of Herzog and Trung on determinantal ideals.