Cesar Ceballos ; Arnau Padrol ; Camilo Sarmiento
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Dyck path triangulations and extendability (extended abstract)
dmtcs:2516 -
Discrete Mathematics & Theoretical Computer Science,
January 1, 2015,
DMTCS Proceedings, 27th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2015)
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https://doi.org/10.46298/dmtcs.2516
Dyck path triangulations and extendability (extended abstract)Conference paper
Authors: Cesar Ceballos 1; Arnau Padrol 2; Camilo Sarmiento 3
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Cesar Ceballos;Arnau Padrol;Camilo Sarmiento
1 Department of Mathematics and Statistics [Toronto]
2 Institut für Mathematik
3 Institut fuer Algebra und Geometrie, Magdeburg
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 Δrn−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 wheneverm≥k>n, any triangulations of Δ(k−1)m−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 interpretations of our results in the language of tropical oriented matroids, which are analogous to classical results in oriented matroid theory.