{
  "id": "2306.00033",
  "version": "v1",
  "published": "2023-05-31T07:32:14.000Z",
  "updated": "2023-05-31T07:32:14.000Z",
  "title": "Sign-Balanced Pattern-Avoiding Permutation Classes",
  "authors": [
    "Junyao Pan",
    "Pengfei Guo"
  ],
  "categories": [
    "math.CO",
    "math.GR"
  ],
  "abstract": "A set of permutations is called sign-balanced if the set contains the same number of even permutations as odd permutations. Let $S_n(\\sigma_1, \\sigma_2, \\ldots, \\sigma_r)$ be the set of permutations in the symmetric group $S_n$ which avoids patterns $\\sigma_1, \\sigma_2, \\ldots, \\sigma_r$. The aim of this paper is to investigate when, for certain patterns $\\sigma_1, \\sigma_2, \\ldots, \\sigma_r$, $S_n(\\sigma_1, \\sigma_2, \\ldots, \\sigma_r)$ is sign-balanced for every integer $n>1$. We prove that for any $\\{\\sigma_1, \\sigma_2, \\ldots, \\sigma_r\\}\\subseteq S_3$, if $\\{\\sigma_1, \\sigma_2, \\ldots, \\sigma_r\\}$ is sign-balanced except $\\{132, 213, 231, 312\\}$, then $S_n(\\sigma_1, \\sigma_2, \\ldots, \\sigma_r)$ is sign-balanced for every integer $n>1$. In addition, we give some results in the case of avoiding some patterns of length $4$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-05-31T07:32:14.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05A05"
    ],
    "keywords": [
      "sign-balanced pattern-avoiding permutation classes",
      "odd permutations",
      "avoids patterns",
      "set contains",
      "symmetric group"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}