  We prove a family of identities, expressing generating functions of powers of
characteristic polynomials of permutations, as finite or infinite products.
These generalize formulae first obtained in a study of the geometry/topology of
symmetric products of real/algebraic tori. The proof uses formal power series
expansions of plethystic exponentials, and has been motivated by some recent
applications of these combinatorial tools in supersymmetric gauge and string
theories. Since the methods are elementary, we tried to be self-contained, and
relate to other topics such as the q-binoomial theorem, and the cycle index and
Molien series for the symmetric group.
