{
  "id": "2102.02920",
  "version": "v1",
  "published": "2021-02-04T22:39:14.000Z",
  "updated": "2021-02-04T22:39:14.000Z",
  "title": "Twenty Vertex model and domino tilings of the Aztec triangle",
  "authors": [
    "Philippe Di Francesco"
  ],
  "comment": "51 pages, 21 figures",
  "categories": [
    "math.CO",
    "cond-mat.stat-mech",
    "math-ph",
    "math.MP"
  ],
  "abstract": "We show that the number of configurations of the 20 Vertex model on certain domains with domain wall type boundary conditions is equal to the number of domino tilings of Aztec-like triangles, proving a conjecture from [P. Di Francesco and E. Guitter, Twenty-Vertex Model with Domain Wall Boundaries and Domino Tilings, Elec. Jour. of Combinatorics 27 (2020), no. 2, P2.13]. The result is based on the integrability of the 20 Vertex model and uses a connection to the U-turn boundary 6 Vertex model to re-express the number of 20 Vertex configurations as a simple determinant, which is then related to a Lindstr\\\"om-Gessel-Viennot determinant for the domino tiling problem. The common number of configurations is conjectured to be $2^{n(n-1)/2}\\prod_{j=0}^{n-1}\\frac{(4j+2)!}{(n+2j+1)!}=1, 4, 60, 3328, 678912...$ The enumeration result is extended to include refinements of both numbers.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-02-04T22:39:14.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "vertex model",
      "domino tiling",
      "aztec triangle",
      "domain wall type boundary conditions",
      "domain wall boundaries"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 51,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}