{
  "id": "math/0610787",
  "version": "v2",
  "published": "2006-10-26T19:13:59.000Z",
  "updated": "2008-01-10T22:55:10.000Z",
  "title": "Shifted set families, degree sequences, and plethysm",
  "authors": [
    "Caroline Klivans",
    "Victor Reiner"
  ],
  "comment": "Final version, 26 pages, 3 figures",
  "categories": [
    "math.CO"
  ],
  "abstract": "We study, in three parts, degree sequences of k-families (or k-uniform hypergraphs) and shifted k-families. The first part collects for the first time in one place, various implications such as: Threshold implies Uniquely Realizable implies Degree-Maximal implies Shifted, which are equivalent concepts for 2-families (=simple graphs), but strict implications for k-families with k > 2. The implication that uniquely realizable implies degree-maximal seems to be new. The second part recalls Merris and Roby's reformulation of the characterization due to Ruch and Gutman for graphical degree sequences and shifted 2-families. It then introduces two generalizations which are characterizations of shifted k-families. The third part recalls the connection between degree sequences of k-families of size m and the plethysm of elementary symmetric functions e_m[e_k]. It then uses highest weight theory to explain how shifted k-families provide the ``top part'' of these plethysm expansions, along with offering a conjecture about a further relation.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2008-01-10T22:55:10.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C07",
      "05C65",
      "05E05"
    ],
    "keywords": [
      "degree sequences",
      "shifted set families",
      "uniquely realizable implies degree-maximal",
      "realizable implies degree-maximal implies"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 26,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2006math.....10787K"
    }
  }
}