The q-analogues of γ-positivity of Eulerian polynomials via group actions
Abstract
Han, Jouhet and Zeng established q-analogues of the γ-expansion formulas for Eulerian polynomials of types A and B. Combinatorial interpretations of the corresponding coefficientsan,k(q) and bn,k(q), however, remained open. In this paper, we provide combinatorial interpretations for thesecoefficients by using the model of increasing binary trees, thereby resolving a problem posed by Han, Jouhet and Zeng. Our combinatorial approach consists of three main steps: 1. construct a Carlitz-type insertion bijection for increasing binary trees and derive a new combinatorial interpretation ofCarlitz's q-Eulerian polynomials of type A in terms of such trees; 2. introduce a generalized Foata--Strehl action on increasing binary trees to interpret the coefficients an,k(q);3. derive a new combinatorial interpretation for the q-Eulerianpolynomials of type B introduced by Chow and Gessel in terms of increasing binary trees of type B, and develop a generalizedFoata--Strehl action on these trees to interpret the coefficients bn,k(q). We further give combinatorial interpretations for the quotients an,k(q)/(−q;q)k−1 and bn,k(q)/(1+q)k(−q;q2)k in terms of André trees and a certain class of increasing binary trees of type B, respectively. As an application of the latter interpretation, we obtain a combinatorial interpretation for a q-analogue of the secant number and prove the positivity conjecture of Han, Jouhet and Zeng.
Related
- The (p,q)-Analogues of Bi-Stirling Eulerian Polynomials via continued fractions
- Classifying Slice-Regular Polynomials via Group Actions on the Twistor Space
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- Real-rooted Eulerian polynomials from permutations, words, and paths
- Multivariate stability of powered triangular recurrences and parametrised Eulerian polynomials
BibTeX
@misc{ding2026qanalogues,
title = {{The $q$-analogues of $γ$-positivity of Eulerian polynomials via group actions}},
author = {Taifeng Ding and Lintong Wang and Sherry H. F. Yan},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.12405},
eprint = {2609.12405},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.12405},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - The q-analogues of γ-positivity of Eulerian polynomials via group actions
AU - Taifeng Ding
AU - Lintong Wang
AU - Sherry H. F. Yan
PY - 2026
DA - 2026/09/11
PB - arXiv
DO - 10.48550/arxiv.2609.12405
UR - https://arxiv.org/abs/2609.12405
AB - Han, Jouhet and Zeng established q-analogues of the γ-expansion formulas for Eulerian polynomials of types A and B. Combinatorial interpretations of the corresponding coefficientsa_n,k(q) and b_n,k(q), however, remained open. In this paper, we provide combinatorial interpretations for thesecoefficients by using the model of increasing binary trees, thereby resolving a problem posed by Han, Jouhet and Zeng. Our combinatorial approach consists of three main steps: 1. construct a Carlitz-type insertion bijection for increasing binary trees and derive a new combinatorial interpretation ofCarlitz's q-Eulerian polynomials of type A in terms of such trees; 2. introduce a generalized Foata--Strehl action on increasing binary trees to interpret the coefficients a_n,k(q);3. derive a new combinatorial interpretation for the q-Eulerianpolynomials of type B introduced by Chow and Gessel in terms of increasing binary trees of type B, and develop a generalizedFoata--Strehl action on these trees to interpret the coefficients b_n,k(q). We further give combinatorial interpretations for the quotients a_n,k(q)/(−q;q)_k−1 and b_n,k(q)/(1+q)^k(−q;q^2)_k in terms of André trees and a certain class of increasing binary trees of type B, respectively. As an application of the latter interpretation, we obtain a combinatorial interpretation for a q-analogue of the secant number and prove the positivity conjecture of Han, Jouhet and Zeng.
ER -