{
  "id": "1409.2745",
  "version": "v1",
  "published": "2014-09-09T14:11:50.000Z",
  "updated": "2014-09-09T14:11:50.000Z",
  "title": "Signed polyomino tilings by n-in-line polyominoes and Groebner bases",
  "authors": [
    "Manuela Muzika Dizdarević",
    "Marinko Timotijević",
    "Rade T. Živaljević"
  ],
  "categories": [
    "math.CO"
  ],
  "abstract": "Conway and Lagarias observed that a triangular region T(m) in a hexagonal lattice admits signed tiling by three-in-line polyominoes (tribones) if and only if m=9d-1 or m=9d for some integer d. We apply the theory of Groebner bases over integers to show that T(m) admits a signed tiling by n-in-line polyominoes (n-bones) if and only if m=dn^2-1 or m=dn^2 for some integer d. Explicit description of the Groebner basis allows us to calculate the \"Groebner discrete volume\" of a lattice region by applying the division algorithm to its `Newton polynomial'. Among immediate consequences is a description of the tile homology group of the $n$-in-line polyomino.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-09-09T14:11:50.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "13P10",
      "05B45"
    ],
    "keywords": [
      "signed polyomino tilings",
      "groebner bases",
      "n-in-line polyominoes",
      "lattice admits signed tiling",
      "groebner discrete volume"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1409.2745M"
    }
  }
}