Polynomial factorization statistics and point configurations in ${\mathbb{R}}^3$

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.