{
  "abstract": "We give a complete rigorous asymptotic analysis of the generating functions of 3-noncrossing skeleton matchings and canonical 3-noncrossing skeleton diagrams. Let $F_3$ be the ordinary generating function of 3-noncrossing matchings, and let $S(y)=\\sum_{n\\geq 0}S(n)y^n$ be determined by $S(zF_3(z)^2)=F_3(z)$. The proof is deliberately ordered to avoid circularity. First, Lagrange inversion, a Stieltjes representation of $F_3$, exact cut-boundary estimates, and a moving horizontal Hankel contour give $S(n)\\sim 24(πA^5)^{-1}σ^{-n}n^{-5}$ independently of any $Δ$-analyticity of $S$. This estimate supplies boundary regularity of $S$ and $S'$. We then prove a global biholomorphic inversion theorem, continuation across every nonprincipal point of the convergence circle, and a logarithmically perturbed sectorial inverse theorem. A complete disk-chain and monodromy argument yields a single-valued continuation to a standard $Δ$-domain. At the principal singularity, $S(y)=Q_4(u)-(πA^5)^{-1}u^4\\log u+O(u^5(1+|\\log u|))$, where $u=1-y/σ$. Finally, the canonical composition $S_3^{[4]}(z)=(1-z)(S(\\vartheta(z))-1-\\vartheta(z))$ is shown to be $Δ$-analytic at its unique dominant singularity $η=0.49340718057613087519\\ldots$, and $[z^n]S_3^{[4]}(z)\\sim 7892.16205625817\\ldots n^{-5}η^{-n}$. The argument retains the methods and detailed estimates of the original proofs while closing the analytic gaps in the earlier dissertation treatment.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.24425",
  "author_total": 1,
  "authors": [
    {
      "name": "Yangyang Zhao"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/45vf0ptv47b4z.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.24425",
  "fulltext_url": null,
  "id": "45vf0ptv47b4z",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.24425",
  "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": "W7204272968",
  "pdf_url": "https://arxiv.org/pdf/2608.24425",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-07",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/45vf0ptv47b4z.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:42Z",
      "native_id": "2608.24425",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.24425"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Rigorous Asymptotic Analysis of 3-Noncrossing Skeleton Diagrams",
  "title_text": "Rigorous Asymptotic Analysis of 3-Noncrossing Skeleton Diagrams",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:42Z",
  "url": "https://research.draconitic.com/paper/45vf0ptv47b4z",
  "venue": "arXiv",
  "versions": []
}