{
  "id": "1602.01126",
  "version": "v1",
  "published": "2016-02-02T21:45:15.000Z",
  "updated": "2016-02-02T21:45:15.000Z",
  "title": "A Combinatorial Approach to the Symmetry of $q,t$-Catalan Numbers",
  "authors": [
    "Kyungyong Lee",
    "Li Li",
    "Nicholas A. Loehr"
  ],
  "categories": [
    "math.CO"
  ],
  "abstract": "The \\emph{$q,t$-Catalan numbers} $C_n(q,t)$ are polynomials in $q$ and $t$ that reduce to the ordinary Catalan numbers when $q=t=1$. These polynomials have important connections to representation theory, algebraic geometry, and symmetric functions. Haglund and Haiman discovered combinatorial formulas for $C_n(q,t)$ as weighted sums of Dyck paths (or equivalently, integer partitions contained in a staircase shape). This paper undertakes a combinatorial investigation of the joint symmetry property $C_n(q,t)=C_n(t,q)$. We conjecture some structural decompositions of Dyck objects into \"mutually opposite\" subcollections that lead to a bijective explanation of joint symmetry in certain cases. A key new idea is the construction of infinite chains of partitions that are independent of $n$ but induce the joint symmetry for all $n$ simultaneously. Using these methods, we prove combinatorially that for $0\\leq k\\leq 9$ and all $n$, the terms in $C_n(q,t)$ of total degree $\\binom{n}{2}-k$ have the required symmetry property.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-02-02T21:45:15.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05A19",
      "05A17",
      "05E05"
    ],
    "keywords": [
      "combinatorial approach",
      "ordinary catalan numbers",
      "joint symmetry property",
      "symmetric functions",
      "combinatorial formulas"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2016arXiv160201126L"
    }
  }
}