{
  "id": "1303.3785",
  "version": "v1",
  "published": "2013-03-15T14:34:37.000Z",
  "updated": "2013-03-15T14:34:37.000Z",
  "title": "The Dyck pattern poset",
  "authors": [
    "Antonio Bernini",
    "Luca Ferrari",
    "Renzo Pinzani",
    "Julian West"
  ],
  "categories": [
    "math.CO"
  ],
  "abstract": "We introduce the notion of pattern in the context of lattice paths, and investigate it in the specific case of Dyck paths. Similarly to the case of permutations, the pattern-containment relation defines a poset structure on the set of all Dyck paths, which we call the Dyck pattern poset. Given a Dyck path P, we determine a formula for the number of Dyck paths covered by P, as well as for the number of Dyck paths covering P. We then address some typical pattern-avoidance issues, enumerating some classes of pattern-avoiding Dyck paths. Finally, we offer a conjecture concerning the asymptotic behavior of the sequence counting Dyck paths avoiding a generic pattern and we pose a series of open problems regarding the structure of the Dyck pattern poset.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2013-03-15T14:34:37.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05A15",
      "06A07"
    ],
    "keywords": [
      "dyck pattern poset",
      "pattern-containment relation defines",
      "specific case",
      "typical pattern-avoidance issues",
      "pattern-avoiding dyck paths"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1303.3785B"
    }
  }
}