Combinatorial Insights into Asymptotic Normality

In this paper we examine various sequences $(X_n)_{n \in \mathbb{N}}$ of real random variables arising naturally in connection with certain combinatorial structures. The goal is to investigate sufficient conditions for convergence to the standard normal distribution upon centering and normalization. By studying signed binomial sums, Stirling numbers of both kinds, and factorial moments, we obtain certain cancellations in the computation of centred moments $\mathbb{E}[(X_n - \mathbb{E}[X_n])^k]$. This allows us to conclude that $\frac{X_n - \mathbb{E}[X_n]}{\sqrt{Var \ X_n}} \xrightarrow{d} N(0,1)$ when the factorial moments $\mathbb{E}[(X_n)_k]$ are rational functions satisfying some simple conditions on the coefficients. A particular class of such random variables are those whose factorial moments are products and quotients of falling factorials in $n$; we apply our result in this situation and we discuss specific examples arising from block statistics of $k$-divisible non-crossing partitions and from outdegree statistics of rooted plane trees with $k$-divisible outdegrees.