We use generating functions to relate the expected values of polynomial factorization statistics over @math to the cohomology of ordered configurations in @math 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.
@cite_2: Etude des partitions en blocs de taille paire et impaire et des caracteres insolites des groupes symetriques. Demonstration d'une conjecture de R.P. Stanley
The first moment formula is derived from the Grothendieck-Lefschetz trace formula for ' e tale cohomology with twisted coefficients'' from which the name is borrowed. The author and Lagarias @cite_2 use Theorem to establish a representation theoretic interpretation of the @math , where @math is the probability of a random squarefree polynomial having factorization type @math .