{
  "id": "1103.1176",
  "version": "v2",
  "published": "2011-03-06T23:31:40.000Z",
  "updated": "2011-11-28T16:33:32.000Z",
  "title": "On the weighted enumeration of alternating sign matrices and descending plane partitions",
  "authors": [
    "Roger E. Behrend",
    "Philippe Di Francesco",
    "Paul Zinn-Justin"
  ],
  "comment": "v2: published version",
  "journal": "J. Combin. Theory Ser. A 119 (2012) 331-363",
  "doi": "10.1016/j.jcta.2011.09.004",
  "categories": [
    "math.CO",
    "math-ph",
    "math.MP"
  ],
  "abstract": "We prove a conjecture of Mills, Robbins and Rumsey [Alternating sign matrices and descending plane partitions, J. Combin. Theory Ser. A 34 (1983), 340-359] that, for any n, k, m and p, the number of nxn alternating sign matrices (ASMs) for which the 1 of the first row is in column k+1 and there are exactly m -1's and m+p inversions is equal to the number of descending plane partitions (DPPs) for which each part is at most n and there are exactly k parts equal to n, m special parts and p nonspecial parts. The proof involves expressing the associated generating functions for ASMs and DPPs with fixed n as determinants of nxn matrices, and using elementary transformations to show that these determinants are equal. The determinants themselves are obtained by standard methods: for ASMs this involves using the Izergin-Korepin formula for the partition function of the six-vertex model with domain-wall boundary conditions, together with a bijection between ASMs and configurations of this model, and for DPPs it involves using the Lindstrom-Gessel-Viennot theorem, together with a bijection between DPPs and certain sets of nonintersecting lattice paths.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2011-11-28T16:33:32.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05A05",
      "05A15",
      "05A19",
      "15B35",
      "82B20",
      "82B23"
    ],
    "keywords": [
      "descending plane partitions",
      "weighted enumeration",
      "determinants",
      "nxn alternating sign matrices",
      "domain-wall boundary conditions"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2011arXiv1103.1176B"
    }
  }
}