M\"obius Inversion and Duality for Summations of Stable Graphs

Using the stratifications of Deligne-Mumford moduli spaces $\overline{\mathcal M}_{g,n}$ indexed by stable graphs, we introduce a partially ordered set of stable graphs by defining a partial ordering on the set of connected stable graphs of genus $g$ with $n$ external edges. By modifying the usual definition of zeta function and M\"obius function of a poset, we introduce generalized ($\mathbb Q$-valued) zeta function and generalized ($\mathbb Q$-valued) M\"obius function of the poset of stable graphs. We use them to proved a generalized M\"obius inversion formula for functions on the poset of stable graphs. Two applications related to duality in earlier work are also presented.