Why? Because Mathics is not up to helping me determine if indeed
f({A1,…,Ap})=(n−p)!2n!p!(1p)n−p|A1|2⋯|Ap|2|A1|!⋯|Ap|!
is a pmf over the set of partitions of the set {1,…,n} into p≤n nonempty sets. In particular, the sets A1,…,Ap in the formula above satisfy |A1|+…+|Ap|=n and |Ai|≥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.