{
  "id": "2107.11715",
  "version": "v1",
  "published": "2021-07-25T03:04:17.000Z",
  "updated": "2021-07-25T03:04:17.000Z",
  "title": "A symmetric chain decomposition of $N(m,n)$ of composition",
  "authors": [
    "Yueming Zhong"
  ],
  "comment": "10 pages, 7 figures",
  "categories": [
    "math.CO"
  ],
  "abstract": "A poset is called a symmetric chain decomposition if the poset can be expressed as a disjoint union of symmetric chains. For positive integers $m$ and $n$, let $N(m,n)$ denote the set of all compositions $\\alpha=(\\alpha_1,\\cdots,\\alpha_m)$, with $0\\le \\alpha_i \\le n$ for each $i=1,\\cdots,m$. Define order $<$ as follow, $\\forall \\alpha,\\beta \\in N(m,n)$, $\\beta < \\alpha$ if and only if $\\beta_i \\le \\alpha_i(i=1,\\cdots,m)$ and $\\sum\\limits_{i=1}^{m}\\beta_i <\\sum\\limits_{i=1}^{m}\\alpha_i$. In this paper, we show that the poset $(N(m,n),<)$ can be expressed as a disjoint of symmetric chains by constructive method.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-07-25T03:04:17.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05A19",
      "05E99"
    ],
    "keywords": [
      "symmetric chain decomposition",
      "disjoint union",
      "define order",
      "positive integers",
      "constructive method"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 10,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}