Parking trees and the toric g-vector of nestohedra

2025/11/06 by Ehrenborg, Richard, Hetyei, Gábor, Readdy, Margaret
Mathematics · Computer Science · #Advanced Combinatorial Mathematics #Polynomial and algebraic computation #Graph theory and applications

paper · doi:10.48550/arxiv.2511.04815

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.

Citations

• On the structure of the d-indivisible noncrossing partition posets
• Labeling regions in deformations of graphical arrangements
• Pattern Avoidance in Parking Functions
• k-Factorizations of the full cycle and generalized Mahonian statistics on k-forests
• Patterns in words of ordered set partitions
• Simion's type B associahedron is a pulling triangulation of the\n Legendre polytope
• Faces of Generalized Permutohedra
• Hard Lefschetz Theorem for Nonrational Polytopes

Related

• Parking functions and triangulation of the associahedron
• Generalized parking functions, descent numbers, and chain polytopes of ribbon posets
• Toric Topology. Chapter 1: Geometry and combinatorics of polytopes
• Tiered tree, Parking function and Postnikov-Shapiro algebra
• H-Representation of the Kimura-3 Polytope