For the Ewens distribution (12.5), show that the conditional distribution given \(\# B=k\) is

\[
p_{\theta}(B \mid \# B=k)=\frac{\prod_{b \in B} \Gamma(\# b)}{s_{n, k}}
\]

where \(s_{n, k}\) is Stirling's number of the first kind, i.e., the number of permutations \([n] ightarrow[n]\) that have exactly \(k\) cycles.