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

Let $K(=K_{n,θ})$ be a positive integer-valued random variable whose distribution is given by ${\rm P}(K = x) = \bar{s}(n,x) θ^x/(θ)_n$ $(x=1,\ldots,n) $, where $θ$ is a positive number, $n$ is a positive integer, $(θ)_n=θ(θ+1)\cdots(θ+n-1)$ and $\bar{s}(n,x)$ is the coefficient of $θ^x$ in $(θ)_n$ for $x=1,\ldots,n$. 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 $θ$ simultaneously tend to infinity, if $n^2/θ\to\infty$, $K$ 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.