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arXiv · 2101.10425

A unifying framework for the $\nu$-Tamari lattice and principal order ideals in Young's lattice

Matias von Bell·Rafael S. González D'León·Francisco A. Mayorga Cetina·Martha Yip
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Abstract

We present a unifying framework in which both the $\nu$-Tamari lattice, introduced by Pr\'eville-Ratelle and Viennot, and principal order ideals in Young's lattice indexed by lattice paths $\nu$, are realized as the dual graphs of two combinatorially striking triangulations of a family of flow polytopes which we call the $\nu$-caracol flow polytopes. The first triangulation gives a new geometric realization of the $\nu$-Tamari complex introduced by Ceballos, Padrol and Sarmiento. We use the second triangulation to show that the $h^*$-vector of the $\nu$-caracol flow polytope is given by the $\nu$-Narayana numbers, extending a result of M\'esz\'aros when $\nu$ is a staircase lattice path. Our work generalizes and unifies results on the dual structure of two subdivisions of a polytope studied by Pitman and Stanley.

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Matias von Bell, Rafael S. González D'León, Francisco A. Mayorga Cetina, Martha Yip. 2021-01-25. A unifying framework for the $\nu$-Tamari lattice and principal order ideals in Young's lattice. https://arxiv.org/abs/2101.10425

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