What is the Möbius function for the lattice of partial partitions?

Question

Let $n$ be a positive integer. Let $P$ be the set of partitions of subsets of $\{ 1, 2, \dotsc, n \}$ (so, for example, when $n=2$, the set $P$ contains $\emptyset$, $\{ \{1 \} \}$, $\{ \{2 \} \}$, $\{ \{1 \}, \{2 \} \}$, and $\{ \{1, 2 \} \}$).

Turn $P$ into a poset by saying $\pi \preceq \sigma$ for $\pi,\sigma \in P$, if every part in $\pi$ is a subset of some part in $\sigma$.

What is the Möbius function of this poset (that is, for every $\pi, \sigma \in P$, I want to know if there is a simple formula which gives the value of $\mu_P(\pi,\sigma)$)?

Answer 1

Hersh, Hanlon, and Shareshian characterize the Möbius function of this lattice in a never-published preprint: A $\operatorname{GL}_n(q)$ analogue of the partition lattice.

It is often 0, but/and agrees up to a sign with that of the (non-partial) partition lattice in the case where the additional elements in the partitioned set of $\sigma$ (over the partitioned set of $\pi$) are all in singletons.