On the $q$-factorization of power series

Any power series with unit constant term can be factored into an infinite product of the form ${\prod }_{n\ge 1}(1-q^n{)}^{-a_n}$. We give direct formulas for the exponents $a_n$ in terms of the coefficients of the power series, and vice versa, as sums over partitions. As examples, we prove identities for certain partition enumeration functions. Finally, we note $q$-analogues of our enumeration formulas.