{
  "id": "1306.5926",
  "version": "v3",
  "published": "2013-06-25T11:39:05.000Z",
  "updated": "2014-04-02T16:48:05.000Z",
  "title": "On the Möbius Function of Permutations With One Descent",
  "authors": [
    "Jason P Smith"
  ],
  "categories": [
    "math.CO"
  ],
  "abstract": "The set of all permutations, ordered by pattern containment, is a poset. We give a formula for the M\\\"obius function of intervals $[1,\\pi]$ in this poset, for any permutation $\\pi$ with at most one descent. We compute the M\\\"obius function as a function of the number and positions of pairs of consecutive letters in $\\pi$ that are consecutive in value. As a result of this we show that the M\\\"obius function is unbounded on the poset of all permutations. We show that the M\\\"obius function is zero on any interval $[1,\\pi]$ where $\\pi$ has a triple of consecutive letters whose values are consecutive and monotone. We also conjecture values of the M\\\"obius function on some other intervals of permutations with at most one descent.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2014-04-02T16:48:05.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05A05"
    ],
    "keywords": [
      "permutation",
      "möbius function",
      "consecutive letters",
      "conjecture values",
      "pattern containment"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1306.5926S"
    }
  }
}