{
  "abstract": "We construct an explicit involution on Dyck paths satisfying $\\defc\\le\\min(\\area,\\dinv)$ that interchanges area and dinv, where $\\defc=\\binom n2-\\area-\\dinv$ and $n$ is the semilength. In addition, we give an involution on Dyck paths with $\\defc \\le 2n-8$ that interchanges area and dinv. Finally, we give an explicit partition formula for the portion of the $q,t$-Catalan polynomial of total degree at least $\\binom n2-2n+8$, thereby proving a conjecture of Lee and Li~\\cite[Conjecture~4]{LeeLi11}. Altogether our results give a combinatorial explanation of $q,t$-Catalan symmetry in the region $\\defc\\le\\max(2n-8,\\min(\\area,\\dinv))$. Our constructions rely on a number of new combinatorial objects developed here and three main external tools: the dual Dyck insertion of \\cite{Hawkes26}, the Garsia--Milne involution principle~\\cite{GarsiaMilne81},~\\cite{Doyle19}, and the partition bijection of ~\\cite{LoehrWarrington09}.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.21403",
  "author_total": 1,
  "authors": [
    {
      "name": "Graham Hawkes"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/3pfgtcxzh1p4a.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.21403",
  "fulltext_url": null,
  "id": "3pfgtcxzh1p4a",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.21403",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CO"
    ],
    [
      "openalex",
      "2607|Advanced Combinatorial Mathematics"
    ]
  ],
  "openalex": "W7213951518",
  "pdf_url": "https://arxiv.org/pdf/2609.21403",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-05",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/3pfgtcxzh1p4a.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:10Z",
      "native_id": "2609.21403",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.21403"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "An involution on Dyck paths in the region $\\defc\\le\\min(\\area,\\dinv)$ that interchanges area and dinv",
  "title_text": "An involution on Dyck paths in the region defc≤min(area,dinv) that interchanges area and dinv",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:10Z",
  "url": "https://research.draconitic.com/paper/3pfgtcxzh1p4a",
  "venue": "arXiv",
  "versions": []
}