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Let $K(=K_{n,\theta})$ be a positive integer-valued random variable whosedistribution is given by ${\rm P}(K = x) = \bar{s}(n,x) \theta^x/(\theta)_n$$(x=1,\ldots,n) $, where $\theta$ is a positive number, $n$ is a positiveinteger, $(\theta)_n=\theta(\theta+1)\cdots(\theta+n-1)$ and $\bar{s}(n,x)$ isthe coefficient of $\theta^x$ in $(\theta)_n$ for $x=1,\ldots,n$. This formuladescribes the distribution of the length of a Ewens partition, which is astandard 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 $n^2/\theta\to\infty$, $K$ alsoasymptotically follows a normal distribution. In this paper, error bounds forthe normal approximation are provided. The result shows that the decay rate ofthe error changes due to asymptotic regimes.

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