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Dyck path triangulations and extendability (extended abstract)

Abstract : 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 $\Delta_{r\ n-1}\times\Delta_{n-1}$ 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 triangulations 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.
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  • HAL Id : hal-01337838, version 1
  • ARXIV : 1402.5111

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Cesar Ceballos, Arnau Padrol, Camilo Sarmiento. Dyck path triangulations and extendability (extended abstract). 27th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2015), Jul 2015, Daejeon, South Korea. pp.73-84. ⟨hal-01337838⟩

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