{
  "id": "2407.07366",
  "version": "v1",
  "published": "2024-07-10T04:54:43.000Z",
  "updated": "2024-07-10T04:54:43.000Z",
  "title": "Counting Permutations in $S_{2n}$ and $S_{2n+1}$",
  "authors": [
    "Yuewen Luo"
  ],
  "categories": [
    "math.CO"
  ],
  "abstract": "Let $\\alpha(n)$ denote the number of perfect square permutations in the symmetric group $S_n$. The conjecture $\\alpha(2n+1) = (2n+1) \\alpha(2n)$, provided by Stanley[4], was proved by Blum[1] using a generating function. This paper presents a combinatorial proof for this conjecture. At the same time, we demonstrate that all permutations with an even number of even cycles in both $S_{2n}$ and $S_{2n+1}$ can be categorized into three distinct types that correspond to each other.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-07-10T04:54:43.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "counting permutations",
      "perfect square permutations",
      "symmetric group",
      "conjecture",
      "combinatorial proof"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}