{
  "id": "1902.08334",
  "version": "v1",
  "published": "2019-02-22T01:58:53.000Z",
  "updated": "2019-02-22T01:58:53.000Z",
  "title": "The Absolute Orders on the Coxeter Groups $A_n$ and $B_n$ are Sperner",
  "authors": [
    "Lawrence H. Harper",
    "Gene B. Kim",
    "Neal Livesay"
  ],
  "comment": "6 pages, 2 tikz figures",
  "categories": [
    "math.CO"
  ],
  "abstract": "Over 50 years ago, Rota posted the following celebrated `Research Problem': prove or disprove that the partial order of partitions on an $n$-set (i.e., the refinement order) is Sperner. A counterexample was eventually discovered by Canfield in 1978. However, Harper and Kim recently proved that a closely related order --- i.e., the refinement order on the symmetric group --- is not only Sperner, but strong Sperner. Equivalently, the well-known absolute order on the symmetric group is strong Sperner. In this paper, we extend these results by giving a concise, elegant proof that the absolute orders on the Coxeter groups $A_n$ and $B_n$ are strong Sperner.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-02-22T01:58:53.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05D05",
      "05E99"
    ],
    "keywords": [
      "coxeter groups",
      "strong sperner",
      "refinement order",
      "symmetric group",
      "well-known absolute order"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 6,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}