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Séminaire Lotharingien de Combinatoire, 82B.50 (2019), 12 pp.

# Emily Barnard and Thomas McConville

# Lattices from graph associahedra

**Abstract.**
Given a graph *G* on *n* vertices, Postnikov defined a graph associahedron *P*_{G} as an example of a generalized permutohedron, a polytope whose normal fan coarsens the braid arrangement. Motivated by two general constructions of subalgebras of the Malvenuto-Reutenauer algebra, we consider the poset *L*_{G} obtained by orienting the one-skeleton of *P*_{G}. Because the normal fan of *P*_{G} coarsens the normal fan of the standard permutohedron we obtain a surjection Ψ_{G}: **S**_{n} -> *L*_{G}. We characterize the graphs *G* for which Ψ_{G} is a lattice quotient map.

Received: November 15, 2018.
Accepted: February 17, 2019.
Final version: April 1, 2019.

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