  We use generating functions to relate the expected values of polynomial
factorization statistics over $\mathbb{F}_q$ to the cohomology of ordered
configurations in $\mathbb{R}^3$ as a representation of the symmetric group.
Our methods lead to a new proof of the twisted Grothendieck-Lefschetz formula
for squarefree polynomial factorization statistics of Church, Ellenberg, and
Farb.
