The toric g-vector of nestohedra

Abstract

We show the entries of the toric g-vector of any dual simplicial polytope is a nonnegative integer linear combination of its gamma-vector entries. We give combinatorial interpretations of the toric g-vector of three classical polytopes: the associahedron, the cyclohedron and the permutahedron. Each is expressed in terms of 123-avoidance on a set of structures, namely, parking functions, functions and parking trees. Using our notion of parking trees, we extend our combinatorial model to all chordal nestohedra. As a corollary we obtain combinatorial proofs of nonnegativity of the toric g-vector for all of these polytopes.