One of the simplest static versions of the Ewens sampling formula is stated as a probability distribution on the set of partitions of the finite set \([n]\) as follows:

\[
P_{n, \alpha}(B)=\frac{\alpha^{\# B} \prod_{b \in B}(\# b-1) !}{\alpha^{\uparrow n}}
\]

where \(\alpha^{\uparrow n}=\alpha(\alpha+1) \cdots(\alpha+n-1)\) is called the rising factorial. Show that \(P_{n, \alpha}\) is an exponential-family model with canonical parameter \(\theta=\log \alpha\), canonical statistic \#B, and cumulant function \(\log \left(\left(e^{\theta}ight)^{\uparrow n}ight)-\log (n !)\).