{
  "id": "2010.06273",
  "version": "v1",
  "published": "2020-10-13T10:32:09.000Z",
  "updated": "2020-10-13T10:32:09.000Z",
  "title": "The feasible regions for consecutive patterns of pattern-avoiding permutations",
  "authors": [
    "Jacopo Borga",
    "Raul Penaguiao"
  ],
  "categories": [
    "math.CO",
    "math.PR"
  ],
  "abstract": "We study the feasible region for consecutive patterns of pattern-avoiding permutations. More precisely, given a family $\\mathcal C$ of permutations avoiding a fixed set of patterns, we study the limit of proportions of consecutive patterns on large permutations of $\\mathcal C$. These limits form a region, which we call the \\emph{pattern-avoiding feasible region for $\\mathcal C$}. We show that, when $\\mathcal C$ is the family of $\\tau$-avoiding permutations, with either $\\tau$ of size three or $\\tau$ a monotone pattern, the pattern-avoiding feasible region for $\\mathcal C$ is a polytope. We also determine its dimension using a new tool for the monotone pattern case whereby we are able to compute the dimension of the image of a polytope after a projection. We further show some general results for the pattern-avoiding feasible region for any family $\\mathcal C$ of permutations avoiding a fixed set of patterns, and we conjecture a general formula for its dimension. Along the way, we discuss connections of this work with the problem of packing patterns in pattern-avoiding permutations and to the study of local limits for pattern-avoiding permutations.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-10-13T10:32:09.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "pattern-avoiding permutations",
      "consecutive patterns",
      "pattern-avoiding feasible region",
      "fixed set",
      "general formula"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}