For some $n \in \mathbb N$, let $(\Pi_n, \le)$ be the poset of partitions of the set $\{1, 2, \dots, n\}$, where two partitions $\pi, \rho \in \Pi_n$ have the relation $\pi \le \rho$ if $\pi$ is a refinement of $\rho$, i.e., if every block in $\pi$ is contained within a block in $\rho$. Möbius inversion on this poset is a fairly standard result in combinatorics: given two functions $f, g: \Pi_n \to \mathbb R$, it is the case that $$ f(\pi) = \sum_{\rho \le \pi} g(\rho), \qquad \forall \pi \in \Pi_n $$ and $$ g(\pi) = \sum_{\rho \le \pi} (-1)^{|\rho| - 1}(|\rho| - 1)! f(\rho), \qquad \forall \pi \in \Pi_n $$ are equivalent. But this result is for lower sums. What is the Möbius inversion formula for upper sums? I.e., if $$ f(\pi) = \sum_{\rho \ge \pi} g(\rho), \qquad \forall \pi \in \Pi_n $$ then what is $g(\pi)$?
UPDATE: I believe I have pieced together an answer, with some help from Martin Aigner's text Combinatorial Theory and some small examples. Given some $\rho = \{C_1, C_2, \dots, C_q\} \in \Pi_n$, there is a one-to-one correspondence between refinements of $\rho$ and the product of partition lattices $\Pi_{|C_1|} \Pi_{|C_2|} \cdots \Pi_{|C_q|}$, since refinements of $\rho$ are obtained by partitioning each block. In other words, we refine $\rho$ by applying a tuple of partitions $(\sigma_1, \sigma_2, \dots, \sigma_q)$ to the blocks $(C_1, C_2, \dots, C_q)$, where $\sigma_C \in \Pi_{|C|}$.
Let us use the notation $\hat 1$ to represent the coarsest (1-block) partition in a lattice $\Pi_k$, and similarly, $\hat 0$ represents the finest ($k$-block) partition. For each block $C \in \rho$, observe that the interval $[\sigma_C, \; \hat 1]$ in $\Pi_{|C|}$ is isomorphic to the interval $[\hat 0, \hat 1]$ in $\Pi_{|\sigma_C|}$. Furthermore, Möbius functions are multiplicative. Combining these observations, we have $$ \mu(\pi, \rho) = \prod_{C \in \rho} \mu_{\Pi_{|\sigma_C|}}(\hat 0, \hat 1) = \prod_{C \in \rho} (-1)^{|\sigma_C| - 1} (|\sigma_C| - 1)! = (-1)^{|\pi| - |\rho|} \prod_{C \in \rho} (|\sigma_C| - 1)! $$ where $(\sigma_1, \sigma_2, \dots, \sigma_q)$ are the partitions that refine $\rho$ down to $\pi$. Then $$ g(\pi) = \sum_{\rho \ge \pi} \mu(\pi, \rho) f(\rho) $$
