{
  "id": "2104.11003",
  "version": "v1",
  "published": "2021-04-22T11:58:28.000Z",
  "updated": "2021-04-22T11:58:28.000Z",
  "title": "An explicit order matching for $L(3,n)$ from several approaches and its extension for $L(4,n)$",
  "authors": [
    "Guoce Xin",
    "Yueming Zhong"
  ],
  "comment": "26 pages, 22 figures",
  "categories": [
    "math.CO"
  ],
  "abstract": "Let $L(m,n)$ denote Young's lattice consisting of all partitions whose Young diagrams are contained in the $m\\times n$ rectangle. It is a well-known result that the poset $L(m,n)$ is rank symmetric, rank unimodal, and Sperner. A direct proof of this result by finding an explicit order matching of $L(m,n)$ is an outstanding open problem. In this paper, we present an explicit order matching $\\varphi$ for $L(3,n)$ by several different approaches, and give chain tableau version of $\\varphi$ that is very helpful in finding patterns. It is surprise that the greedy algorithm and a recursive knead process also give the same order matching. Our methods extend for $L(4,n)$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-04-22T11:58:28.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "explicit order matching",
      "approaches",
      "chain tableau version",
      "young diagrams",
      "well-known result"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 26,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}