{
  "id": "1305.2206",
  "version": "v2",
  "published": "2013-05-09T20:13:43.000Z",
  "updated": "2015-11-24T21:15:05.000Z",
  "title": "Symmetries of statistics on lattice paths between two boundaries",
  "authors": [
    "Sergi Elizalde",
    "Martin Rubey"
  ],
  "comment": "Small typos corrected, and journal reference and grant info added",
  "journal": "Adv. Math. 287 (2016), 347-388",
  "doi": "10.1016/j.aim.2015.09.025",
  "categories": [
    "math.CO"
  ],
  "abstract": "We prove that on the set of lattice paths with steps N=(0,1) and E=(1,0) that lie between two fixed boundaries T and B (which are themselves lattice paths), the statistics `number of E steps shared with B' and `number of E steps shared with T' have a symmetric joint distribution. To do so, we give an involution that switches these statistics, preserves additional parameters, and generalizes to paths that contain steps S=(0,-1) at prescribed x-coordinates. We also show that a similar equidistribution result for path statistics follows from the fact that the Tutte polynomial of a matroid is independent of the order of its ground set. We extend the two theorems to k-tuples of paths between two boundaries, and we give some applications to Dyck paths, generalizing a result of Deutsch, to watermelon configurations, to pattern-avoiding permutations, and to the generalized Tamari lattice. Finally, we prove a conjecture of Nicol\\'as about the distribution of degrees of k consecutive vertices in k-triangulations of a convex n-gon. To achieve this goal, we provide a new statistic-preserving bijection between certain k-tuples of non-crossing paths and k-flagged semistandard Young tableaux, which is based on local moves reminiscent of jeu de taquin.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2013-05-09T20:13:43.000Z",
      "abstract": "We prove that on the set of lattice paths with steps N=(0,1) and E=(1,0) that lie between two boundaries T and B, the statistics 'number of E steps shared with B' and 'number of E steps shared with T' have a symmetric joint distribution. We give an involution that switches these statistics, preserves additional parameters, and generalizes to paths that contain steps S=(0,-1) at prescribed x-coordinates. We also show that a similar equidistribution result for path statistics follows from the fact that the Tutte polynomial of a matroid is independent of the order of its ground set. We extend the two theorems to k-tuples of paths between two boundaries, and we give some applications to Dyck paths, generalizing a result of Deutsch, to watermelon configurations, and to pattern-avoiding permutations. Finally, we prove a conjecture of Nicol\\'as about the distribution of degrees of k consecutive vertices in a k-triangulation of a convex n-gon, by providing a new bijection between certain k-tuples of paths and k-flagged semistandard Young tableaux.",
      "comment": null,
      "journal": null,
      "doi": null
    },
    {
      "version": "v2",
      "updated": "2015-11-24T21:15:05.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05A19",
      "52C05",
      "05A15",
      "05B35",
      "05E05"
    ],
    "keywords": [
      "lattice paths",
      "boundaries",
      "symmetries",
      "similar equidistribution result",
      "preserves additional parameters"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "publisher": "Elsevier",
      "journal": "Adv. Math."
    },
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1305.2206E"
    }
  }
}