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Séminaire Lotharingien de Combinatoire, 85B.42 (2021), 12 pp.

# Matias von Bell, Rafael S. González D'León, Francisco A. Mayorga Cetina and Martha Yip

# On Framed Triangulations of Flow Polytopes, the ν-Tamari Lattice and Young's Lattice

**Abstract.**
We study two combinatorially striking triangulations of a family of flow polytopes indexed by lattice paths ν which we call the ν-caracol flow polytopes. The first triangulation gives a geometric realization of the ν-Tamari complex introduced by Ceballos, Padrol and Sarmiento, whose dual graph is the Hasse diagram of the ν-Tamari lattice introduced by Préville-Ratelle and Viennot.
The dual graph of the second triangulation is the Hasse diagram of the principal order ideal determined by ν in Young's lattice.
We use the latter triangulation to show that the *h*^{*}-vector of the ν-caracol flow polytope is given by the ν-Narayana numbers, extending the result of Mészáros when ν is a staircase lattice path.

Received: December 1, 2020.
Accepted: March 1, 2021.
Final version: April 29, 2021.

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