Dyck path triangulations and extendability

We introduce the Dyck path triangulation of the cartesian product of two simplices Δn−1×Δn−1Δn−1×Δn−1. The maximal simplices of this triangulation are given by Dyck paths, and the construction naturally generalizes to certain rational Dyck paths. Our study of the Dyck path triangulation is motivated by extendability problems of partial triangulations of products of two simplices. We show that whenever m≥k>nm≥k>n, any triangulation of the product of the k  -skeleton of Δm−1Δm−1 with Δn−1Δn−1 extends to a unique triangulation of Δm−1×Δn−1Δm−1×Δn−1. Moreover, using the Dyck path triangulation, we prove that the bound k>nk>n is optimal. We also exhibit interpretations of our results in the language of tropical oriented matroids that are analogous to classical results in oriented matroid theory.