{
  "id": "2408.15075",
  "version": "v1",
  "published": "2024-08-27T14:03:55.000Z",
  "updated": "2024-08-27T14:03:55.000Z",
  "title": "Enumerating 1324-avoiders with few inversions",
  "authors": [
    "Svante Linusson",
    "Emil Verkama"
  ],
  "comment": "23 pages",
  "categories": [
    "math.CO"
  ],
  "abstract": "We enumerate the numbers $Av_n^k(1324)$ of 1324-avoiding $n$-permutations with exactly $k$ inversions for all $k$ and $n \\geq (k+7)/2$. The result depends on a structural characterization of such permutations in terms of a new notion of almost-decomposability. In particular, our enumeration verifies half of a conjecture of Claesson, Jel\\'inek and Steingr\\'imsson, according to which $Av_n^k(1324) \\leq Av_{n+1}^k(1324)$ for all $n$ and $k$. Proving also the other half would improve the best known upper bound for the exponential growth rate of the number of $1324$-avoiders from $13.5$ to approximately $13.002$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-08-27T14:03:55.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05A05",
      "05A15"
    ],
    "keywords": [
      "inversions",
      "exponential growth rate",
      "enumeration verifies half",
      "enumerating",
      "structural characterization"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 23,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}