{
  "id": "1106.0952",
  "version": "v3",
  "published": "2011-06-06T03:02:00.000Z",
  "updated": "2011-06-17T15:30:20.000Z",
  "title": "A Combinatorial Formula for Rank 2 Cluster Variables",
  "authors": [
    "Kyungyong Lee",
    "Ralf Schiffler"
  ],
  "comment": "17 pages, v2:a corollary to the main theorem added, v3:another corollary added",
  "categories": [
    "math.CO",
    "math.RT"
  ],
  "abstract": "Let $r$ be any positive integer, and let $x_1, x_2$ be indeterminates. We consider the sequence $\\{x_n\\}$ defined by the recursive relation $$ x_{n+1} =(x_n^r +1)/{x_{n-1}} $$ for any integer $n$. Finding a combinatorial expression for $x_n$ as a rational function of $x_1$ and $x_2$ has been an open problem since 2001. We give a direct elementary formula for $x_n$ in terms of subpaths of a specific lattice path in the plane. The formula is manifestly positive, providing a new proof of a result by Nakajima and Qin.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2011-06-17T15:30:20.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "13F60",
      "05C70"
    ],
    "keywords": [
      "cluster variables",
      "combinatorial formula",
      "specific lattice path",
      "direct elementary formula",
      "open problem"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 17,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2011arXiv1106.0952L"
    }
  }
}