{
  "id": "1312.5758",
  "version": "v2",
  "published": "2013-12-19T21:28:42.000Z",
  "updated": "2014-08-17T18:45:09.000Z",
  "title": "A Distributive Lattice Connected with Arithmetic Progressions of Length Three",
  "authors": [
    "Fu Liu",
    "Richard P. Stanley"
  ],
  "comment": "25 pages, 1 figure. To appear in the Ramanujan Journal",
  "categories": [
    "math.CO"
  ],
  "abstract": "Let $\\mathcal{T}$ be a collection of 3-element subsets $S$ of $\\{1, \\ldots,n\\}$ with the property that if $i<j<k$ and $a<b<c$ are two 3-element subsets in $S$, then there exists an integer sequence $x_1 < x_2 < \\cdots < x_n$ such that $x_i, x_j, x_k$ and $x_a, x_b, x_c$ are arithmetic progressions. We determine the number of such collections $\\mathcal{T}$ and the number of them of maximum size. These results confirm two conjectures of Noam Elkies.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2014-08-17T18:45:09.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05A15"
    ],
    "keywords": [
      "arithmetic progressions",
      "distributive lattice",
      "collection",
      "integer sequence",
      "results confirm"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 25,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1312.5758L"
    }
  }
}