= Factorial moment densities of Poisson-Dirichlet components
{title2=$\alpha_\theta^{(m)}(dx)=\theta^m\frac{(1-\sum_i x_i)^{\theta-1}}{\prod_i x_i}\,dx$}
This <probability density function>[density] is supported where all $x_i>0$ and $\sum_i x_i<1$, and gives the expected sum over ordered distinct components of $PD(\theta)$. Apply the iterated <Mecke formula for a Poisson random measure> to $m$ selected jumps $u_i$ and the remaining gamma total $t$. With $s=t+\sum_i u_i$ and $x_i=u_i/s$, the <Jacobian determinant>[Jacobian] is $s^m$ and the <integral> in $s$ is $\Gamma(\theta)$, giving the <probability density function>[density]. It describes joint counts of large components, beyond the first-moment <measure>.
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