{
  "id": "math/0409562",
  "version": "v3",
  "published": "2004-09-28T20:57:17.000Z",
  "updated": "2005-08-04T02:59:54.000Z",
  "title": "On Stanley's reciprocity theorem for rational cones",
  "authors": [
    "Matthias Beck",
    "Mike Develin"
  ],
  "comment": "9 pages",
  "categories": [
    "math.CO"
  ],
  "abstract": "We give a short, self-contained proof of Stanley's reciprocity theorem for a rational cone K \\subset R^d. Namely, let sigma_K (x) = sum_{m \\in K \\cap Z^d} x^m. Then sigma_K (x) and sigma_int(K) (x) are rational functions which satisfy the identity sigma_K (1/x) = (-1)^d sigma_int(K) (x). A corollary of Stanley's theorem is the Ehrhart-Macdonald reciprocity theorem for the lattice-point enumerator of rational polytopes. A distinguishing feature of our proof is that it uses neither the shelling of a polyhedron nor the concept of finite additive measures. The proof follows from elementary techniques in contour integration.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2005-08-04T02:59:54.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05A15",
      "52C07"
    ],
    "keywords": [
      "stanleys reciprocity theorem",
      "rational cone",
      "ehrhart-macdonald reciprocity theorem",
      "stanleys theorem",
      "lattice-point enumerator"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 9,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2004math......9562B"
    }
  }
}