Parking trees and the toric g-vector of nestohedra

TL;DR

The paper establishes a general formula expressing the toric $g$-vector $g(P,x)$ of a simple polytope as a nonnegative combination of its $\gamma$-vector entries, enabling combinatorial interpretations for classical polytopes. It primes a series of results: for the associahedron, the toric $g$-vector entries equal ascent counts in $123$-avoiding parking functions; for the cyclohedron, analogous counts arise for $123$-avoiding functions; and for the permutahedron, the entries enumerate ascents in parking trees representing $123$-avoiding parking functions. The framework extends to all chordal nestohedra, linking toric $g$-polynomials to restricted Foata--Strehl actions and parking-tree encodings, and it culminates in a cohesive combinatorial landscape connecting polytopal invariants with classical counting paradigms. The authors also pose open problems about real-rootedness and simplicial realizations, suggesting rich directions for future work in algebraic combinatorics and polyhedral geometry.

Abstract

We express the toric g-vector entries of any simple polytope as a nonnegative integer linear combination of its gamma-vector entries. Using this expression we obtain 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.

Figures (5)

• Figure 1: The Krattenthaler Dyck path $w = U^{4} D^{2} U^{2} D^{3} U^{3} D^{2} U D^{3}$ for the $123$-avoiding permutation $\pi=(\underline{7},10,\underline{5},9,8,\underline{2},6,\underline{1},4,3)$ where the left-to-right minima are underlined.
• Figure 2: The Dyck path of Garsia and Haiman representing the parking function $f = (7,5,10,7,3,6,1,4,3,1)$. The associated permutation is $\pi = (\underline{7},10,\underline{5},9,8,\underline{2},6,\underline{1},4,3)$ and the Dyck word is $v = U^{2} D^{2} U^{2} D U D U D U D U^{2} D^{3} U D$.
• Figure 3: The Foata--Strehl trees for the permutations $\tau = (2$, $1$, $4$, $5$, $6$, $8$, $7$, $9$, $3$, $10$, $11)$, $\phi_{5}\tau$ and $\phi_{1}\phi_{5}\tau$.
• Figure 4: The pair of Dyck paths representing the parking function $(7,5,10,7,3,6,1,4,3,1)$.
• Figure 5: The depth-first search and breadth-first search parking trees of the same parking function $(7,5,10,7,3,6,1,4,3,1)$.

Theorems & Definitions (84)

• Conjecture 1.1
• Lemma 2.1
• Definition 2.2
• Definition 2.3
• Lemma 2.4
• Definition 2.5
• Definition 2.6
• Definition 2.7
• Definition 2.8
• Definition 2.9