{
  "id": "2001.09123",
  "version": "v1",
  "published": "2020-01-24T18:14:10.000Z",
  "updated": "2020-01-24T18:14:10.000Z",
  "title": "What fraction of an $S_n$-orbit can lie on a hyperplane?",
  "authors": [
    "Jiahui Huang",
    "David McKinnon",
    "Matthew Satriano"
  ],
  "comment": "16 pages",
  "categories": [
    "math.CO"
  ],
  "abstract": "Consider the $S_n$-action on $\\mathbb{R}^n$ given by permuting coordinates. This paper addresses the following problem: compute $\\max_{v,H} |H\\cap S_nv|$ as $H\\subset\\mathbb{R}^n$ ranges over all hyperplanes through the origin and $v\\in\\mathbb{R}^n$ ranges over all vectors with distinct coordinates that are not contained in the hyperplane $\\sum x_i=0$. We conjecture that for $n\\geq3$, the answer is $(n-1)!$ for odd $n$, and $n(n-2)!$ for even $n$. We prove that if $p$ is the largest prime with $p\\leq n$, then $\\max_{v,H} |H\\cap S_nv|\\leq \\frac{n!}{p}$. In particular, this proves the conjecture when $n$ or $n-1$ is prime.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-01-24T18:14:10.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05E99",
      "05A05"
    ],
    "keywords": [
      "hyperplane",
      "largest prime",
      "conjecture",
      "distinct coordinates",
      "paper addresses"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 16,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}