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Séminaire Lotharingien de Combinatoire, 78B.68 (2017), 12 pp.

# Cesar Ceballos, Arnau Padrol and Carmilo Sarmiento

#
Geometry of ν-Tamari Lattices in Types *A* and *B*

**Abstract.**
In this extended abstract, we exploit the combinatorics and geometry
of triangulations of products of simplices to reinterpret and
generalize a number of constructions in Catalan combinatorics. In our
framework, the main role of "Catalan objects" is played by
(*I*,*J*^{-})-trees: bipartite trees associated to a pair
(*I*,*J*^{-})
of finite index sets that stand in simple bijection
with lattice paths weakly above a lattice path
ν = ν(*I*,*J*^{-}).
Such trees label the maximal simplices of a
triangulation of a subpolytope of the cartesian product of two
simplices, which provides a geometric realization of the ν-Tamari
lattice introduced by Préville-Ratelle and Viennot. Dualizing this
triangulation, we obtain a polyhedral complex induced by an
arrangement of tropical hyperplanes whose 1-skeleton realizes the
Hasse diagram of the ν-Tamari lattice, and thus generalizes the
simple associahedron. Specializing to the Fuß-Catalan case realizes
the *m*-Tamari lattices as 1-skeleta of regular subdivisions of
classical associahedra, giving a positive answer to a question of
F. Bergeron. The simplicial complex underlying our triangulation has
its *h*-vector given by a suitable generalization of the Narayana
numbers. We propose it as a natural generalization of the classical
simplicial associahedron, alternative to the rational associahedron of
Armstrong, Rhoades and Williams.
Our methods are amenable to cyclic symmetry, which we use to present
type-*B* analogues of our constructions. Notably, we define a partial
order that generalizes the type *B* Tamari lattice, introduced
independently by Thomas and Reading, along with corresponding
geometric realizations.

Received: November 14, 2016.
Accepted: February 17, 2017.
Final version: April 1, 2017.

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