{
  "id": "0710.5014",
  "version": "v2",
  "published": "2007-10-26T07:47:21.000Z",
  "updated": "2007-11-15T09:21:29.000Z",
  "title": "On $k$-noncrossing partitions",
  "authors": [
    "Emma Y. Jin",
    "Jing Qin",
    "Christian M. Reidys"
  ],
  "comment": "5 pages; 3 figures",
  "categories": [
    "math.CO",
    "math.RT"
  ],
  "abstract": "In this paper we prove a duality between $k$-noncrossing partitions over $[n]=\\{1,...,n\\}$ and $k$-noncrossing braids over $[n-1]$. This duality is derived directly via (generalized) vacillating tableaux which are in correspondence to tangled-diagrams \\cite{Reidys:07vac}. We give a combinatorial interpretation of the bijection in terms of the contraction of arcs of tangled-diagrams. Furthermore it induces by restriction a bijection between $k$-noncrossing, 2-regular partitions over $[n]$ and $k$-noncrossing braids without isolated points over $[n-1]$. Since braids without isolated points correspond to enhanced partitions this allows, using the results of \\cite{MIRXIN}, to enumerate 2-regular, 3-noncrossing partitions.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2007-11-15T09:21:29.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "06A07"
    ],
    "keywords": [
      "noncrossing partitions",
      "noncrossing braids",
      "isolated points correspond",
      "tangled-diagrams",
      "combinatorial interpretation"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 5,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2007arXiv0710.5014J"
    }
  }
}