{
  "abstract": "The study of $t$-intersecting families in symmetric groups received lots of attention in the last two decades. In this paper, we study two different kinds of stability results for $t$-intersecting families in symmetric groups. Let $Σ_n$ denote the symmetric group on $\\{1,2,\\ldots,n\\}$. The $t$-diversity $γ_t(\\mathcal{F})$ of $\\mathcal{F}\\subseteqΣ_n$ is defined as the minimum number of members of $\\mathcal{F}$ whose deletion results in a family with transversal number $t$. The star $t$-diversity $γ_t^{\\star}(\\mathcal{F})$ is defined as the minimum number of members of $\\mathcal{F}$ whose deletion results in a $t$-star. For $n$ relatively large with respect to $t$, we determine the best possible bounds for both $γ_t(\\mathcal{F})$ and $γ_t^{\\star}(\\mathcal{F})$ over all $t$-intersecting families $\\mathcal{F}\\subseteqΣ_n$. The equality holding conditions are characterized. For $γ_t(\\mathcal{F})$, the extremal case is generated by all $2t$-subsets of a $3t$-partial permutation or, when $t=2$, the family complements of the lines of the Fano plane. For $γ_t^{\\star}(\\mathcal{F})$, the extremal family is generated by all $(t+1)$-subsets of a $(t+2)$-partial permutation.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.12391",
  "author_total": 5,
  "authors": [
    {
      "name": "Yuhang Cao"
    },
    {
      "name": "Gennian Ge"
    },
    {
      "name": "Jian Wang"
    },
    {
      "name": "Jialuo Wang"
    },
    {
      "name": "Xiaochen Zhao"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/fezj7d8fypvpy.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.12391",
  "fulltext_url": null,
  "id": "fezj7d8fypvpy",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.12391",
  "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": "W7213172445",
  "pdf_url": "https://arxiv.org/pdf/2609.12391",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-11",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/fezj7d8fypvpy.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:21:11Z",
      "native_id": "2609.12391",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.12391"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "The maximum $t$-diversity of $t$-intersecting families of permutations",
  "title_text": "The maximum t-diversity of t-intersecting families of permutations",
  "type": "preprint",
  "updated_at": "2026-10-30T12:21:11Z",
  "url": "https://research.draconitic.com/paper/fezj7d8fypvpy",
  "venue": "arXiv",
  "versions": []
}