{
  "id": "1808.03764",
  "version": "v1",
  "published": "2018-08-11T07:31:52.000Z",
  "updated": "2018-08-11T07:31:52.000Z",
  "title": "Restricted permutations refined by number of crossings and nestings",
  "authors": [
    "Paul M. Rakotomamonjy"
  ],
  "comment": "17 pages, 5 tables and 7 figures",
  "categories": [
    "math.CO"
  ],
  "abstract": "Let $st=\\{st_1,st_2,...,st_k\\}$ be a set of k statistics on permutations with $k\\geq 1$. We say that two given subset of permutations $T$ and $T'$ are $st$-Wilf-equivalent if the joint distributions of all statistics in st over the sets of $T$-avoiding permutations $S_n(T)$ and $T'$-avoiding permutations $S_n(T')$ are the same. The main purpose of this paper is the $(cr,nes)$-Wilf-equivalence classes for all single pattern in $S_3$, where cr and nes denote respectively the statistics number of crossings and nestings. One of the main tools that we use is the bijection $\\Theta$ from the set of 321-avoiding permutations $S_n(321)$ to the set of 132-avoiding permutations $S_n(132)$ which was exhibited by Elizalde and Pak. They proved that the bijection $\\Theta$ preserves the number of fixed points and excedances. Since the given formulation of $\\Theta$ is not direct, we show that it can be defined directly by a recursive formula. Then, we prove that it also preserves the number of crossings. These properties of the bijection $\\Theta$ leads to an unexpected result related to the q,p-Catalan numbers defined by Randrianarivony.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-08-11T07:31:52.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05A19",
      "05A20",
      "05A05"
    ],
    "keywords": [
      "restricted permutations",
      "avoiding permutations",
      "joint distributions",
      "wilf-equivalence classes",
      "p-catalan numbers"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 17,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}