I have a question related to this one. For $n,p \in \mathbb{N}_+$ such that $p\mid n$, let $\mathcal{P}^{\rm eq}$ be the set of all *equipartitions* of $n$ in $p$ sets; i.e., in sets of equal size $\frac{n}{p}$.

Is it known how to sample efficiently (i.e., in time polynomial in $n,p$) from the uniform distribution on $\mathcal{P}^{\rm eq}$?

**Edit:** I had forgotten a key part: can this be done using the optimal (up to constant factors) number of random bits, i.e., $O(n\log p)$ uniformly random bits?