{
  "id": "1506.03249",
  "version": "v1",
  "published": "2015-06-10T10:37:26.000Z",
  "updated": "2015-06-10T10:37:26.000Z",
  "title": "q-Stirling numbers: A new view",
  "authors": [
    "Yue Cai",
    "Margaret A. Readdy"
  ],
  "categories": [
    "math.CO"
  ],
  "abstract": "We show the classical $q$-Stirling numbers of the second kind can be expressed more compactly as a pair of statistics on a subset of restricted growth words. The resulting expressions are polynomials in $q$ and $1+q$. We extend this enumerative result via a decomposition of a new poset $\\Pi(n,k)$ which we call the Stirling poset of the second kind. Its rank generating function is the $q$-Stirling number $S_q[n,k]$. The Stirling poset of the second kind supports an algebraic complex and a basis for integer homology is determined. A parallel enumerative, poset theoretic and homological study for the $q$-Stirling numbers of the first kind is done. Letting $t = 1+q$ we give a bijective argument showing the $(q,t)$-Stirling numbers of the first and second kind are orthogonal.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-06-10T10:37:26.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05A18",
      "05A30",
      "06A07",
      "11B73",
      "18G35"
    ],
    "keywords": [
      "q-stirling numbers",
      "second kind supports",
      "stirling poset",
      "rank generating function",
      "algebraic complex"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv150603249C"
    }
  }
}