{
  "id": "1708.05188",
  "version": "v1",
  "published": "2017-08-17T09:54:04.000Z",
  "updated": "2017-08-17T09:54:04.000Z",
  "title": "Asymptotics for a Class of Meandric Systems, via the Hasse Diagram of NC(n)",
  "authors": [
    "I. P. Goulden",
    "Alexandru Nica",
    "Doron Puder"
  ],
  "comment": "35 pages, 9 Figures",
  "categories": [
    "math.CO",
    "math.OA",
    "math.PR"
  ],
  "abstract": "We consider closed meandric systems, and their equivalent description in terms of the Hasse diagrams of the lattices of non-crossing partitions $NC(n)$. In this equivalent description, considerations on the number of components of a random meandric system of order $n$ translate into considerations about the distance between two random partitions in $NC(n)$. We focus on a class of couples $(\\pi,\\rho)\\in NC(n)^2$ -- namely the ones where $\\pi$ is conditioned to be an interval partition -- for which it turns out to be tractable to study distances in the Hasse diagram. As a consequence, we observe a non-trivial class of meanders (i.e. connected meandric systems), which we call \"meanders with shallow top\", and which can be explicitly enumerated. Moreover, the expected number of components for a random \"meandric system with shallow top\", is asymptotically $(9n+28)/27$. A variation of the methods used in the shallow-top case yields non-trivial bounds on the expected number of components of a general (unconditioned) random meandric system of order $n$. We show this expected number falls inside $(0.17n,0.51n)$ for large enough $n$. Our calculations here are related to the idea of taking the derivative at $t=1$ in a semigroup for the operation $\\boxplus$ of free probability (but the underlying considerations are presented in a self-contained way). Another variation of these methods goes by fixing a \"base-point\" $\\lambda_{n}$, where $\\lambda_{n}$ is an interval partition in $NC(n)$, and by focusing on distances in the Hasse diagram of $NC(n)$ which are measured from $\\lambda_{n}$. We illustrate this by determining precise formulas for the average distance to $\\lambda_{n}$ and for the cardinality of $\\{\\rho\\in NC(n)\\mid\\rho$ at maximal distance from $\\lambda_{n}\\}$ in the case when $n$ is even and $\\lambda_{n}$ is the partition with blocks $\\{1,2\\},\\{3,4\\},\\ldots,\\{n-1,n\\}$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-08-17T09:54:04.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05A15",
      "05C10",
      "46L54",
      "52C30",
      "05A18",
      "05A16"
    ],
    "keywords": [
      "hasse diagram",
      "random meandric system",
      "expected number",
      "shallow-top case yields non-trivial bounds",
      "interval partition"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 35,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}