{
  "abstract": "Improper edges were introduced by Shor to refine Cayley's formula for rooted labeled trees. Zeng established a connection between Shor's refinement and the Ramanujan polynomials. Let $\\mathscr{T}_n$ denote the set of rooted labeled trees on $[n]=\\{1,\\ldots,n\\}$. We prove that the number of improper edges in a uniformly random tree in $\\mathscr{T}_n$ is asymptotically normal as $n\\to\\infty$, with mean and variance asymptotic to $μn$ and $σ^2 n$, respectively, where $μ=e-2$ and $σ^2=e^2-3e+1$. This phenomenon was observed by Chen, and the proof presented here was developed through human--AI collaboration.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.26045",
  "author_total": 2,
  "authors": [
    {
      "name": "Wennie W. J. Ma"
    },
    {
      "name": "Kathy Q. Ji"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/9ka3smq1xv4m8.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.26045",
  "fulltext_url": null,
  "id": "9ka3smq1xv4m8",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.26045",
  "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": "W7214105873",
  "pdf_url": "https://arxiv.org/pdf/2609.26045",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-22",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/9ka3smq1xv4m8.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:43Z",
      "native_id": "2609.26045",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.26045"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "On the limiting distribution of the number of improper edges for random trees",
  "title_text": "On the limiting distribution of the number of improper edges for random trees",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:43Z",
  "url": "https://research.draconitic.com/paper/9ka3smq1xv4m8",
  "venue": "arXiv",
  "versions": []
}