Error bounds for the normal approximation to the length of a Ewens partition

Let $K(=K_{n,\theta })$ be a positive integer-valued random variable whose distribution is given by $\mathrm{P}(K=x)=\bar{s}(n,x){\theta }^x\mathrm{/}(\theta {)}_n$ $(x=1,\dots ,n)$, where $\theta$ is a positive number, $n$ is a positive integer, $(\theta {)}_n=\theta (\theta +1)\cdots (\theta +n-1)$ and $\bar{s}(n,x)$ is the coefficient of ${\theta }^x$ in $(\theta {)}_n$ for $x=1,\dots ,n$. This formula describes the distribution of the length of a Ewens partition, which is a standard model of random partitions. As $n$ tends to infinity, $K$ asymptotically follows a normal distribution. Moreover, as $n$ and $\theta$ simultaneously tend to infinity, if $n^2\mathrm{/}\theta \to \mathrm{\infty }$, $K$ also asymptotically follows a normal distribution. In this paper, error bounds for the normal approximation are provided. The result shows that the decay rate of the error changes due to asymptotic regimes.