Let @math denote the number of monic irreducible polynomials in @math of degree @math . We show that for a fixed degree @math , the sequence @math converges @math -adically to an explicitly determined rational function @math . Furthermore we show that the limit @math is related to the classic necklace polynomial @math by an involutive functional equation, leading to a phenomenon we call liminal reciprocity. The limiting first moments of factorization statistics for squarefree polynomials are expressed in terms of a family of symmetric group representations as a consequence of liminal reciprocity
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.