{
  "id": "1907.10123",
  "version": "v1",
  "published": "2019-07-23T20:46:56.000Z",
  "updated": "2019-07-23T20:46:56.000Z",
  "title": "Trees, Parking Functions and Factorizations of Full Cycles",
  "authors": [
    "John Irving",
    "Amarpreet Rattan"
  ],
  "comment": "23 pages, 8 figures",
  "categories": [
    "math.CO"
  ],
  "abstract": "Parking functions of length $n$ are well known to be in correspondence with both labelled trees on $n+1$ vertices and factorizations of the full cycle $\\sigma_n=(0\\,1\\,\\cdots\\,n)$ into $n$ transpositions. In fact, these correspondences can be refined: Kreweras equated the area enumerator of parking functions with the inversion enumerator of labelled trees, while an elegant bijection of Stanley maps the area of parking functions to a natural statistic on factorizations of $\\sigma_n$. We extend these relationships in two principal ways. First, we introduce a bivariate refinement of the inversion enumerator of trees and show that it matches a similarly refined enumerator for factorizations. Secondly, we characterize all full cycles $\\sigma$ such that Stanley's function remains a bijection when the canonical cycle $\\sigma_n$ is replaced by $\\sigma$. We also exhibit a connection between our refined inversion enumerator and Haglund's bounce statistic on parking functions.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-07-23T20:46:56.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05A15"
    ],
    "keywords": [
      "parking functions",
      "full cycle",
      "factorizations",
      "labelled trees",
      "haglunds bounce statistic"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 23,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}