{
  "id": "1110.3566",
  "version": "v2",
  "published": "2011-10-17T03:03:20.000Z",
  "updated": "2012-09-20T08:40:29.000Z",
  "title": "Asymptotics of the number of threshold functions on a two-dimensional rectangular grid",
  "authors": [
    "Pentti Haukkanen",
    "Jorma K. Merikoski"
  ],
  "categories": [
    "math.CO",
    "cs.IT",
    "math.IT",
    "math.LO",
    "math.NT"
  ],
  "abstract": "Let $m,n\\ge 2$, $m\\le n$. It is well-known that the number of (two-dimensional) threshold functions on an $m\\times n$ rectangular grid is {eqnarray*} t(m,n)=\\frac{6}{\\pi^2}(mn)^2+O(m^2n\\log{n})+O(mn^2\\log{\\log{n}})= \\frac{6}{\\pi^2}(mn)^2+O(mn^2\\log{m}). {eqnarray*} We improve the error term by showing that $$ t(m,n)=\\frac{6}{\\pi^2}(mn)^2+O(mn^2). $$",
  "revisions": [
    {
      "version": "v2",
      "updated": "2012-09-20T08:40:29.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "03B50",
      "05A99",
      "11N37",
      "11P21"
    ],
    "keywords": [
      "two-dimensional rectangular grid",
      "threshold functions",
      "asymptotics"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2011arXiv1110.3566H"
    }
  }
}