{
  "id": "2404.08470",
  "version": "v1",
  "published": "2024-04-12T13:39:56.000Z",
  "updated": "2024-04-12T13:39:56.000Z",
  "title": "Variations of the $α$-Eulerian polynomials and gamma positivity",
  "authors": [
    "Chao Xu",
    "Jiang Zeng"
  ],
  "comment": "19 pages",
  "categories": [
    "math.CO"
  ],
  "abstract": "We prove a connection formula between two multivariate generalizations of the Eulerian polynomials $A^{\\cyc}_n(x,y, t\\,|\\,\\alpha)$ and $A_n(u_1,u_2,u_3,u_4, f, g, t\\,|\\,\\alpha, \\beta)$, which enumerate permutations related to excedance and descent based statistics respectively. By exploring this connection, we derive the exponential generating function of the latter polynomials and several $\\gamma$-positivity formulas for variants of Eulerian polynomials. In particular, our results generalise the main results in two recent papers by Ji \\cite{Ji23} and Ji-Lin \\cite{JL23}. Our proofs are combinatorial in nature and involve Foata's fundamental transformation and a cyclic analogue of valley-hopping.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-04-12T13:39:56.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05A15",
      "05A19"
    ],
    "keywords": [
      "eulerian polynomials",
      "gamma positivity",
      "variations",
      "foatas fundamental transformation",
      "multivariate generalizations"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 19,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}