{
  "id": "1111.5736",
  "version": "v1",
  "published": "2011-11-24T12:08:13.000Z",
  "updated": "2011-11-24T12:08:13.000Z",
  "title": "Upper bounds for the Stanley-Wilf limit of 1324 and other layered patterns",
  "authors": [
    "Anders Claesson",
    "Vít Jelínek",
    "Einar Steingrímsson"
  ],
  "journal": "J. Comb. Theory A, 119(8) (2012), 1680-1691",
  "doi": "10.1016/j.jcta.2012.05.006",
  "categories": [
    "math.CO"
  ],
  "abstract": "We prove that the Stanley-Wilf limit of any layered permutation pattern of length $\\ell$ is at most $4\\ell^2$, and that the Stanley-Wilf limit of the pattern 1324 is at most 16. These bounds follow from a more general result showing that a permutation avoiding a pattern of a special form is a merge of two permutations, each of which avoids a smaller pattern. If the conjecture is true that the maximum Stanley-Wilf limit for patterns of length $\\ell$ is attained by a layered pattern then this implies an upper bound of $4\\ell^2$ for the Stanley-Wilf limit of any pattern of length $\\ell$. We also conjecture that, for any $k\\ge 0$, the set of 1324-avoiding permutations with $k$ inversions contains at least as many permutations of length $n+1$ as those of length $n$. We show that if this is true then the Stanley-Wilf limit for 1324 is at most $e^{\\pi\\sqrt{2/3}} \\simeq 13.001954$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2011-11-24T12:08:13.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "upper bound",
      "layered pattern",
      "maximum stanley-wilf limit",
      "conjecture",
      "special form"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2011arXiv1111.5736C"
    }
  }
}