{
  "abstract": "Two permutations $σ,τ\\in S_n$ are called disjoint if the composition $στ^{-1}$ has no fixed point. If a family $\\mathcal F\\subseteq S_n$ contains no $s$ pairwise disjoint permutations, then a simple averaging argument gives $|\\mathcal F|\\leq(s-1)(n-1)!$. Inozemtsev, Kolupaev and Kupavskii characterized the equality cases in the range $s\\leq n/(2^{17}\\log n).$ We characterize all equality cases throughout the range $2\\le s\\le n$: equality holds if and only if $\\mathcal F$ is a union of $(s-1)$ pairwise disjoint $1$-cosets. We also prove the linear statement underlying this classification: a real-valued function on $S_n$ has constant sum on every one-factorization if and only if it lies in the span of the indicators of the $1$-cosets. The proof is combinatorial and applies to every order, with a few small orders handled separately.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.03904",
  "author_total": 2,
  "authors": [
    {
      "name": "Mengyu Cao"
    },
    {
      "name": "Haixiang Zhang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/1yz4pdzhd3mhj.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.03904",
  "fulltext_url": null,
  "id": "1yz4pdzhd3mhj",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.03904",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CO"
    ],
    [
      "openalex",
      "2607|Advanced Combinatorial Mathematics"
    ]
  ],
  "openalex": "W7208758922",
  "pdf_url": "https://arxiv.org/pdf/2609.03904",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-03",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/1yz4pdzhd3mhj.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:18:36Z",
      "native_id": "2609.03904",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.03904"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Extremal Families for Matchings in Permutations",
  "title_text": "Extremal Families for Matchings in Permutations",
  "type": "preprint",
  "updated_at": "2026-10-30T12:18:36Z",
  "url": "https://research.draconitic.com/paper/1yz4pdzhd3mhj",
  "venue": "arXiv",
  "versions": []
}