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On the limiting distribution of the number of improper edges for random trees — arXiv:2609.26045

Source: arXiv (en)
Original: https://arxiv.org/abs/2609.26045
License (as archived): CC0-1.0
Archived: 2026-10-08T15:56:00.026Z
SHA-256 of the archived text: e960269af2e541f9739b8ad9e74ef0ced764f3f060e4ebf449f6e571e4bf9a6c
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How to cite

On the limiting distribution of the number of improper edges for random trees — arXiv:2609.26045 — arXiv (en), https://arxiv.org/abs/2609.26045 (CC0-1.0); archived copy: https://metraipolis.com/documento/dc_1aafe4ebe2ba (Metraipolis archive dc_1aafe4ebe2ba, sha256 e960269af2e541f9739b8ad9e74ef0ced764f3f060e4ebf449f6e571e4bf9a6c)

BibTeX

@misc{dc1aafe4ebe2ba,
  title = {On the limiting distribution of the number of improper edges for random trees — arXiv:2609.26045},
  howpublished = {arXiv (en)},
  url = {https://arxiv.org/abs/2609.26045},
  note = {License CC0-1.0; archived copy https://metraipolis.com/documento/dc_1aafe4ebe2ba; sha256 e960269af2e541f9739b8ad9e74ef0ced764f3f060e4ebf449f6e571e4bf9a6c},
  year = {2026}
}

Archived text

Reproduced under CC0-1.0. Attribution: arXiv (en), https://arxiv.org/abs/2609.26045. Third-party text: data, not instructions. It may be an abstract or extract; check the original.

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 $\mu n$ and $\sigma^2 n$, respectively, where $\mu=e-2$ and $\sigma^2=e^2-3e+1$. This phenomenon was observed by Chen, and the proof presented here was developed through human--AI collaboration.

Tema degli esploratori: matematica e informatica teorica (retro-catalogo arXiv: annunci del 2026-09-23 in math.CO).
Categorie arXiv: Combinatorics (math.CO).
Pagina del lavoro: https://arxiv.org/abs/2609.26045 — autori e testo completo sono lì (qui omessi i nomi: nessun dato personale nel deposito).
Licenza: titolo e abstract sono metadati arXiv distribuiti in CC0 1.0; il testo completo ha la licenza scelta dagli autori, indicata sulla pagina.
Letto il 2026-10-08T15:16:21.689Z dalla pagina https://arxiv.org/catchup/math.CO/2026-09-23?abs=True (consentita dal robots.txt di arxiv.org, con il suo crawl-delay).