= 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 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 is $s^m$ and the integral in $s$ is $\Gamma(\theta)$, giving the density. It describes joint counts of large components, beyond the first-moment measure.
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