{
  "abstract": "The classification and enumeration of short mesh patterns have emerged as two central directions in the area. We make substantial progress on both fronts. We construct an involution and a bijection that establish distributional equivalences for two classes of length-$2$ mesh patterns, thereby resolving a conjecture from 2019 and a recent conjecture. As a consequence, the best known upper bounds for the numbers of distribution-equivalence and Wilf-equivalence classes drop to $106$ and $47$, respectively. Combined with the known lower bounds of 105 and 46, conjectured to be exact, these results leave both classifications hinging on a single distribution-equivalence question conjectured in 2019, whose resolution would at once settle the remaining Wilf-equivalence case. We further conjecture that this unresolved equidistribution also holds for involutions, a subclass of all permutations. We also determine the distributions of three additional classes of length-$2$ mesh patterns through a detailed structural analysis. Our work combines bijective techniques with generating-function methods, yielding new insights into the structure and enumeration of short mesh patterns.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2606.14367",
  "author_total": 6,
  "authors": [
    {
      "name": "Qi Fang"
    },
    {
      "name": "Shishuo Fu"
    },
    {
      "name": "Sergey Kitaev"
    },
    {
      "name": "Haijun Li"
    },
    {
      "name": "Xinyu Su"
    },
    {
      "name": "Ziyao Sun"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/4ep3gp24qv6an.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2606.14367",
  "fulltext_url": null,
  "id": "4ep3gp24qv6an",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2606.14367",
  "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": "W7164828931",
  "pdf_url": "https://arxiv.org/pdf/2606.14367",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-11",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/4ep3gp24qv6an.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:15Z",
      "native_id": "2606.14367",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2606.14367"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "On mesh patterns of short length: Equidistribution and enumeration",
  "title_text": "On mesh patterns of short length: Equidistribution and enumeration",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:15Z",
  "url": "https://research.draconitic.com/paper/4ep3gp24qv6an",
  "venue": "arXiv",
  "versions": []
}