Generalized triangulations and diagonal-free subsets of stack polyominoes

For n ⩾ 3 , let Ω n 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 Ω n such that all 2 j endpoints are distinct. For k ⩾ 1 , let Δ n , k be the simplicial complex of subsets of Ω n not containing any ( k + 1 ) -crossing. For example, Δ n , 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 Δ n , k have the same number k ( 2 n - 2 k - 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.