Generalized triangulations and diagonal-free subsets of stack polyominoes

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 distinct 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,1 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 is 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 generalize 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.