{
  "id": "0710.1816",
  "version": "v1",
  "published": "2007-10-09T16:11:54.000Z",
  "updated": "2007-10-09T16:11:54.000Z",
  "title": "Crossings and Nestings of Two Edges in Set Partitions",
  "authors": [
    "Svetlana Poznanovik",
    "Catherine Yan"
  ],
  "comment": "19 pages, 2 figures",
  "categories": [
    "math.CO"
  ],
  "abstract": "Let $\\pi$ and $\\lambda$ be two set partitions with the same number of blocks. Assume $\\pi$ is a partition of $[n]$. For any integer $l, m \\geq 0$, let $\\mathcal{T}(\\pi, l)$ be the set of partitions of $[n+l]$ whose restrictions to the last $n$ elements are isomorphic to $\\pi$, and $\\mathcal{T}(\\pi, l, m)$ the subset of $\\mathcal{T}(\\pi,l)$ consisting of those partitions with exactly $m$ blocks. Similarly define $\\mathcal{T}(\\lambda, l)$ and $\\mathcal{T}(\\lambda, l,m)$. We prove that if the statistic $cr$ ($ne$), the number of crossings (nestings) of two edges, coincides on the sets $\\mathcal{T}(\\pi, l)$ and $\\mathcal{T}(\\lambda, l)$ for $l =0, 1$, then it coincides on $\\mathcal{T}(\\pi, l,m)$ and $\\mathcal{T}(\\lambda, l,m)$ for all $l, m \\geq 0$. These results extend the ones obtained by Klazar on the distribution of crossings and nestings for matchings.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2007-10-09T16:11:54.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05A18",
      "05A15"
    ],
    "keywords": [
      "set partitions",
      "results extend",
      "restrictions",
      "similarly define",
      "distribution"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 19,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2007arXiv0710.1816P"
    }
  }
}