{
  "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 coefficients$a_{n,k}(q)$ 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.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.12405",
  "author_total": 3,
  "authors": [
    {
      "name": "Taifeng Ding"
    },
    {
      "name": "Lintong Wang"
    },
    {
      "name": "Sherry H. F. Yan"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/awjgzp2tzv9rh.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.12405",
  "fulltext_url": null,
  "id": "awjgzp2tzv9rh",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.12405",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CO"
    ],
    [
      "openalex",
      "2607|Advanced Combinatorial Mathematics"
    ]
  ],
  "openalex": "W7212653362",
  "pdf_url": "https://arxiv.org/pdf/2609.12405",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-11",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/awjgzp2tzv9rh.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:21:11Z",
      "native_id": "2609.12405",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.12405"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "The $q$-analogues of $γ$-positivity of Eulerian polynomials via group actions",
  "title_text": "The q-analogues of γ-positivity of Eulerian polynomials via group actions",
  "type": "preprint",
  "updated_at": "2026-10-30T12:21:11Z",
  "url": "https://research.draconitic.com/paper/awjgzp2tzv9rh",
  "venue": "arXiv",
  "versions": []
}