{
  "id": "1708.05518",
  "version": "v1",
  "published": "2017-08-18T06:51:29.000Z",
  "updated": "2017-08-18T06:51:29.000Z",
  "title": "Signed Countings of types B and D permutations and $t,q$-Euler Numbers",
  "authors": [
    "Sen-Peng Eu",
    "Tung-Shan Fu",
    "Hsiang-Chun Hsu",
    "Hsin-Chieh Liao"
  ],
  "categories": [
    "math.CO"
  ],
  "abstract": "It is a classical result that the parity-balance of the number of weak excedances of all permutations (derangements, respectively) of length $n$ is the Euler number $E_n$, alternating in sign, if $n$ is odd (even, respectively). Josuat-Verg\\`{e}s obtained a $q$-analog of the results respecting the number of crossings of a permutation. One of the goals in this paper is to extend the results to the permutations (derangements, respectively) of types B and D, on the basis of the joint distribution in statistics excedances, crossings and the number of negative entries obtained by Corteel, Josuat-Verg\\`{e}s and Kim. Springer numbers are analogous Euler numbers that count the alternating permutations of type B, called snakes. Josuat-Verg\\`{e}s derived bivariate polynomials $Q_n(t,q)$ and $R_n(t,q)$ as generalized Euler numbers via successive $q$-derivatives and multiplications by $t$ on polynomials in $t$. The other goal in this paper is to give a combinatorial interpretation of $Q_n(t,q)$ and $R_n(t,q)$ as the enumerators of the snakes with restrictions.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-08-18T06:51:29.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "permutation",
      "signed countings",
      "weak excedances",
      "joint distribution",
      "derangements"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}