{
  "id": "2006.14064",
  "version": "v1",
  "published": "2020-06-24T21:32:52.000Z",
  "updated": "2020-06-24T21:32:52.000Z",
  "title": "Derivatives, Eulerian polynomials and the $g$-indexes of Young tableaux",
  "authors": [
    "G. -N. Han",
    "S. -M. Ma"
  ],
  "categories": [
    "math.CO"
  ],
  "abstract": "In this paper we first present summation formulas for $k$-order Eulerian polynomials and $1/k$-Eulerian polynomials. We then present combinatorial expansions of $(c(x)D)^n$ in terms of inversion sequences as well as $k$-Young tableaux, where $c(x)$ is a differentiable function in the indeterminate $x$ and $D$ is the derivative with respect to $x$. We define the $g$-indexes of $k$-Young tableaux and Young tableaux, which have important applications in combinatorics. By establishing some relations between $k$-Young tableaux and standard Young tableaux, we express Eulerian polynomials, second-order Eulerian polynomials, Andr\\'e polynomials and the generating polynomials of gamma coefficients of Eulerian polynomials in terms of standard Young tableaux, which imply a deep connection among these polynomials.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-06-24T21:32:52.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05A05",
      "05A17"
    ],
    "keywords": [
      "standard young tableaux",
      "derivative",
      "second-order eulerian polynomials",
      "express eulerian polynomials",
      "summation formulas"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}