Do–Galvin conjecture on asymptotic normality of graph Stirling sequences for acyclic graphs

For each $n\geq 0$, let $G_n$ be an acyclic graph on $n$ vertices, and write ${G_n \brace k}$ for the number of partitions of the vertex set of $G_n$ into $k$ non-empty independent sets. A sequence $(a_{n,k})_{k\in\mathbb Z}$ is asymptotically normal if, for the random variable $X_n$ with

$$\Pr(X_n=k)=\frac{a_{n,k}}{\sum_{j\geq 0}a_{n,j}},$$

we have, uniformly in $x\in\mathbb R$,

$$\Pr\left(\frac{X_n-\mu_n}{\sigma_n}\leq x\right)\longrightarrow \Pr(Z\leq x)$$

as $n\to\infty$, where $\mu_n$ and $\sigma_n$ are the mean and standard deviation of $X_n$, and $Z$ is standard normal. Do–Galvin conjecture. If $G_n$ is an acyclic graph on $n$ vertices, then the graph Stirling sequence

$$\left({G_n \brace k}\right)_{k\in\mathbb Z}$$

is asymptotically normal. This extends Harper's theorem for the empty graphs, whose graph Stirling sequences are the ordinary Stirling sequences of the second kind. Do and Galvin proved the conjecture under the additional condition that $G_n$ has no more than $o\!\left(\sqrt{n/\log n}\right)$ components; the unrestricted acyclic case remains open.