Dyck path triangulations and extendability

We introduce the Dyck path triangulation of the cartesian product of two simplices ${\mathrm{\Delta }}_{n-1}\times {\mathrm{\Delta }}_{n-1}$. The maximal simplices of this triangulation are given by Dyck paths, and its construction naturally generalizes to produce triangulations of ${\mathrm{\Delta }}_{rn-1}\times {\mathrm{\Delta }}_{n-1}$ 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 $m\ge k>n$, any triangulation of ${\mathrm{\Delta }}_{m-1}^{(k-1)}\times {\mathrm{\Delta }}_{n-1}$ extends to a unique triangulation of ${\mathrm{\Delta }}_{m-1}\times {\mathrm{\Delta }}_{n-1}$. 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.