{
  "id": "1208.3831",
  "version": "v3",
  "published": "2012-08-19T12:57:08.000Z",
  "updated": "2013-07-19T21:20:45.000Z",
  "title": "The $\\s$-Eulerian polynomials have only real roots",
  "authors": [
    "Carla D. Savage",
    "Mirkó Visontai"
  ],
  "comment": "27 pages, revised version",
  "categories": [
    "math.CO"
  ],
  "abstract": "We study the roots of generalized Eulerian polynomials via a novel approach. We interpret Eulerian polynomials as the generating polynomials of a statistic over inversion sequences. Inversion sequences (also known as Lehmer codes or subexcedant functions) were recently generalized by Savage and Schuster, to arbitrary sequences $\\s$ of positive integers, which they called $\\s$-inversion sequences. Our object of study is the generating polynomial of the {\\em ascent} statistic over the set of $\\s$-inversion sequences of length $n$. Since this ascent statistic over inversion sequences is equidistributed with the descent statistic over permutations we call this generalized polynomial the \\emph{$\\s$-Eulerian polynomial}. The main result of this paper is that, for any sequence $\\s$ of positive integers, the $\\s$-Eulerian polynomial has only real roots. This result is first shown to generalize many existing results about the real-rootedness of various Eulerian polynomials. We then show that it can be used to settle a conjecture of Brenti, that Eulerian polynomials for all finite Coxeter groups have only real roots. It is then extended to several $q$-analogs. We also show that the MacMahon--Carlitz $q$-Eulerian polynomial has only real roots whenever $q$ is a positive real number confirming a conjecture of Chow and Gessel. The same holds true for the $(\\des,\\finv)$-generating polynomials and also for the $(\\des,\\fmaj)$-generating polynomials for the hyperoctahedral group and the wreath product groups, confirming further conjectures of Chow and Gessel, and Chow and Mansour, respectively.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2013-07-19T21:20:45.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05A05",
      "26C10"
    ],
    "keywords": [
      "real roots",
      "inversion sequences",
      "generating polynomial",
      "interpret eulerian polynomials",
      "positive integers"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 27,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1208.3831S"
    }
  }
}