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

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 $$\mu_P(\pi,\sigma)$$)?

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.