{
  "id": "1201.4767",
  "version": "v3",
  "published": "2012-01-23T16:58:50.000Z",
  "updated": "2013-02-02T00:00:38.000Z",
  "title": "Shape-Wilf-equivalences for vincular patterns",
  "authors": [
    "Andrew M. Baxter"
  ],
  "comment": "15 pages, 7 figures, 1 table. Presented at Permutation Patterns 2012; Accepted to Advanced in Applied Mathematics",
  "doi": "10.1016/j.aam.2013.01.003",
  "categories": [
    "math.CO"
  ],
  "abstract": "We extend the notion of shape-Wilf-equivalence to vincular patterns (also known as \"generalized patterns\" or \"dashed patterns\"). First we introduce a stronger equivalence on patterns which we call filling-shape-Wilf-equivalence. When vincular patterns $\\alpha$ and $\\beta$ are filling-shape-Wilf-equivalent, we prove that the direct sum $\\alpha\\oplus\\sigma$ is filling-shape-Wilf-equivalent to $\\beta\\oplus\\sigma$. We also discover two new pairs of patterns which are filling-shape-Wilf-equivalent: when $\\alpha$, $\\beta$, and $\\sigma$ are nonempty consecutive patterns which are Wilf-equivalent, $\\alpha\\oplus\\sigma$ is filling-shape-Wilf-equivalent to $\\beta\\oplus\\sigma$; and for any consecutive pattern $\\alpha$, $1\\oplus\\alpha$ is filling-shape-Wilf-equivalent to $1\\ominus\\alpha$. These equivalences generalize Wilf-equivalences found by Elizalde and Kitaev. These new equivalences imply many new Wilf-equivalences for vincular patterns",
  "revisions": [
    {
      "version": "v3",
      "updated": "2013-02-02T00:00:38.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05A05",
      "05A15",
      "05A19"
    ],
    "keywords": [
      "vincular patterns",
      "filling-shape-wilf-equivalent",
      "direct sum",
      "nonempty consecutive patterns",
      "stronger equivalence"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 15,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1201.4767B"
    }
  }
}