{
  "id": "math/0508112",
  "version": "v1",
  "published": "2005-08-05T20:19:40.000Z",
  "updated": "2005-08-05T20:19:40.000Z",
  "title": "A Refinement of the Eulerian Numbers, and the Joint Distribution of $π(1)$ and Des($π$) in $S_n$",
  "authors": [
    "Mark Conger"
  ],
  "comment": "21 pages, 4 figures",
  "categories": [
    "math.CO",
    "math.PR"
  ],
  "abstract": "Given a permutation $\\pi$ chosen uniformly from $S_n$, we explore the joint distribution of $\\pi(1)$ and the number of descents in $\\pi$. We obtain a formula for the number of permutations with $\\des(\\pi)=d$ and $\\pi(1)=k$, and use it to show that if $\\des(\\pi)$ is fixed at $d$, then the expected value of $\\pi(1)$ is $d+1$. We go on to derive generating functions for the joint distribution, show that it is unimodal if viewed correctly, and show that when $d$ is small the distribution of $\\pi(1)$ among the permutations with $d$ descents is approximately geometric. Applications to Stein's method and the Neggers-Stanley problem are presented.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2005-08-05T20:19:40.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05A15",
      "20B30",
      "60C05"
    ],
    "keywords": [
      "joint distribution",
      "eulerian numbers",
      "refinement",
      "permutation",
      "steins method"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 21,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2005math......8112C"
    }
  }
}