随机树中非正常边数量的极限分布
On the limiting distribution of the number of improper edges for random trees
- Tianjin University（天津大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文证明有根标记随机树中非正常边数量渐近正态，均值与方差分别渐近于$(e-2)n$和$(e^2-3e+1)n$，该证明由人机协作完成。
AI中文摘要：
非正常边由Shor引入，用以细化有根标记树的Cayley公式。Zeng建立了Shor细化与Ramanujan多项式之间的联系。设$\mathscr{T}_n$表示$[n]=\{1,\ldots,n\}$上有根标记树的集合。我们证明，当$n\to\infty$时，$\mathscr{T}_n$中均匀随机树的非正常边数量渐近正态，其均值与方差分别渐近于$\mu n$和$\sigma^2 n$，其中$\mu=e-2$，$\sigma^2=e^2-3e+1$。这一现象由Chen观察到，本文给出的证明是通过人机协作开发的。
英文摘要：
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.
