{
  "id": "0709.3810",
  "version": "v3",
  "published": "2007-09-24T16:44:30.000Z",
  "updated": "2008-08-21T18:24:38.000Z",
  "title": "Asymptotic estimates for the number of contingency tables, integer flows, and volumes of transportation polytopes",
  "authors": [
    "Alexander Barvinok"
  ],
  "comment": "35 pages, estimates sharpened, details and references added",
  "categories": [
    "math.CO",
    "math.MG"
  ],
  "abstract": "We prove an asymptotic estimate for the number of mxn non-negative integer matrices (contingency tables) with prescribed row and column sums and, more generally, for the number of integer feasible flows in a network. Similarly, we estimate the volume of the polytope of mxn non-negative real matrices with prescribed row and column sums. Our estimates are solutions of convex optimization problems and hence can be computed efficiently. As a corollary, we show that if row sums R=(r_1, ..., r_m) and column sums C=(c_1, ..., c_n) with r_1 + ... + r_m =c_1 + ... +c_n =N are sufficiently far from constant vectors, then, asymptotically, in the uniform probability space of the mxn non-negative integer matrices with the total sum N of entries, the event consisting of the matrices with row sums R and the event consisting of the matrices with column sums C are positively correlated.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2008-08-21T18:24:38.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05A16",
      "60C05",
      "52A38",
      "52B12",
      "52B55"
    ],
    "keywords": [
      "asymptotic estimate",
      "contingency tables",
      "transportation polytopes",
      "column sums",
      "integer flows"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 35,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2007arXiv0709.3810B"
    }
  }
}