Why? Because Mathics is not up to helping me determine if indeed
\[
f(\{A_1, \ldots, A_p\}) = \frac{(n-p)!^2}{n! p!} \left(\frac{1}{p} \right)^{n-p} \frac{|A_1|^2 \cdots |A_p|^2}{|A_1|!\cdots |A_p|!}
\]
is a pmf over the set of partitions of the set \(\{1, \ldots, n\}\) into \(p \leq n\) nonempty sets. In particular, the sets \(A_1, \ldots, A_p\) in the formula above satisfy \(|A_1| + \ldots +|A_p| = n\) and \(|A_i| \geq 1\) for all \(i.\)
I feel like I could empirically test this easily in Mathematica, but OMG trying to do it in Matlab is a real pain, so I gave up. Combinatorics or set manipulation in Matlab in general is an exercise in trying to make a smoothie with a grater: you can do it, eventually, but it’s going to take forever and make a mess.