{
  "id": "1402.0825",
  "version": "v6",
  "published": "2014-02-04T18:42:40.000Z",
  "updated": "2015-04-01T16:00:56.000Z",
  "title": "Generating function of the tilings of Aztec rectangle with holes",
  "authors": [
    "Tri Lai"
  ],
  "comment": "15 pages",
  "categories": [
    "math.CO"
  ],
  "abstract": "We consider a generating function of the domino tilings of an Aztec rectangle with several boundary unit squares removed. Our generating function involves two statistics: the rank of the tiling and half number of vertical dominoes as in the Aztec diamond theorem by Elkies, Kuperberg, Larsen and Propp. In addition, our work deduces a combinatorial explanation for an interesting connection between the number of lozenge tilings of a semihexagon and the number of domino tilings of an Aztec rectangle.",
  "revisions": [
    {
      "version": "v5",
      "updated": "2014-03-17T20:30:46.000Z",
      "title": "Weighted Aztec diamond graphs revisited",
      "abstract": "Kamioka (Journal of Combinatorial Theory, Series A, 2014) considered a certain weighted Aztec diamond graph when presenting a new proof for Aztec diamond theorem by Elkies, Kuperberg, Larsen and Propp. In this paper, we generalize Kamioka's result by investigating several new weighted Aztec diamond graphs. We also prove a generalization for a result due to Mills, Robbins, Rumsey on holey Aztec rectangle. The result implies a weighted version of MacMahon's theorem on rhombus tilings. In addition, we prove simple product formulas for the numbers of perfect matching of new families of weighted graphs.",
      "comment": "44 pages",
      "journal": null,
      "doi": null
    },
    {
      "version": "v6",
      "updated": "2015-04-01T16:00:56.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05A15",
      "05E99",
      "05B45"
    ],
    "keywords": [
      "weighted aztec diamond graphs",
      "simple product formulas",
      "aztec diamond theorem",
      "holey aztec rectangle",
      "generalize kamiokas result"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 15,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1402.0825L"
    }
  }
}