Summary: We introduce the Dyck path triangulation of the Cartesian product of two simplices \(\Delta_{n-1}\times\Delta_{n-1}\). The maximal simplices of this triangulation are given by Dyck paths, and its construction naturally generalizes to produce triangulations of \(\Delta_{rn-1}\times\Delta_{n-1}\) 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 \(m\geq k> n\), any triangulation of \(\Delta_{m-1}^{(k-1)}\times\Delta_{n-1}\) extends to a unique triangulation of \(\Delta_{m-1}\times\Delta_{n-1}\). Moreover, with an explicit construction, we prove that the bound \(k > n\) is optimal. We also exhibit interpretations of our results in the language of tropical oriented matroids, which are analogous to classical results in oriented matroid theory.
For the entire collection see [Zbl 1333.05004].