Parking trees and the toric g-vector of nestohedra

TL;DR

This paper links the toric g-vector of certain polytopes like associahedra, cyclohedra, and permutahedra to ascent statistics of parking functions and trees, providing combinatorial interpretations and extending to chordal nestohedra.

Contribution

It introduces a new combinatorial interpretation of the toric g-vector entries using parking functions and trees, extending to all chordal nestohedra.

Findings

Toric g-vector of associahedron equals ascent statistic of 123-avoiding parking functions

Toric g-vector of cyclohedron relates to 123-avoiding functions

Toric g-vector of permutahedron records ascent statistics of parking trees

Abstract

We express the toric g-vector entries of any simple polytope as a nonnegative integer linear combination of its gamma-vector entries. We show that the toric g-vector of the associahedron is the ascent statistic of 123-avoiding parking functions. An analogous result holds for the cyclohedron and 123-avoiding functions. We prove that the toric g-vector of the permutahedron records the ascent statistics of parking trees representing 123-avoiding parking functions. We indicate how our approach extends to all chordal nestohedra.