Dyck path triangulations and extendability

We introduce the Dyck path triangulation of the cartesian product of two simplices ∆n−1×∆n−1. The maximal simplices of this triangulation are given by Dyck paths, and its construction naturally generalizes to produce triangulations of ∆rn−1 × ∆n−1 using 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 > n, any triangulation of ∆ m−1 ×∆n−1 extends to a unique triangulation of ∆m−1 × ∆n−1. Moreover, with an explicit construction, we prove that the bound k > n is optimal. We also exhibit interesting interpretations of our results in the language of tropical oriented matroids, which are analogous to classical results in oriented matroid theory.