Year: 2026
Type: preprint
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.
Source: arXiv (Cornell University)
Authors: Ma, Wennie W. J., Ji, Kathy Q.
Institution: Tianjin University
Language: English
Subfield: Discrete Mathematics and Combinatorics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7214105873