2014/02/20 by Cesar Ceballos, Ceballos, Cesar, Arnau Padrol +3
Computer Science · Mathematics · #Advanced Combinatorial Mathematics #Combinatorics (math.CO) #Commutative Algebra and Its Applications #FOS: Mathematics #Polynomial and algebraic computation
paper · doi:10.48550/arxiv.1402.5111
openalex publication_date 2014/02/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We introduce the Dyck path triangulation of the cartesian product of two simplices Δn-1×Δn-1. The maximal simplices of this triangulation are given by Dyck paths, and its construction naturally generalizes to produce triangulations of Δr n-1×Δn-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 Δm-1(k-1)×Δn-1 extends to a unique triangulation of Δm-1×Δn-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.