We develop a calculus for $S_n$-equivariant Euler characteristics of moduli spaces of stable curves and stable maps. Our approach involves an enrichment of Pólya's cycle index polynomial of a graph to a certain algebra $Λ^{[2]}$ of wreath product symmetric functions. Building on foundational work of Macdonald, we prove that $Λ^{[2]}$ may be viewed as the Grothendieck ring of the category of polynomial functors which map symmetric sequences of vector spaces to vector spaces. This interpretation gives rise to an action of $Λ^{[2]}$ on the ordinary ring of symmetric functions $Λ$, which is described concretely in terms of Adams operations and skewing by power sums. This action lets us deduce appealing formulas, involving only ordinary symmetric functions, for generating functions of $S_n$-equivariant Euler characteristics.