  Suppose that for each n >= 0 we have a representation $M_n$ of the symmetric
group S_n. Such sequences arise in a wide variety of contexts, and often
exhibit uniformity in some way. We prove a number of general results along
these lines in this paper: our prototypical theorem states that if $M_n$ can be
given a suitable module structure over a twisted commutative algebra then the
sequence $M_n$ follows a predictable pattern. We phrase these results precisely
in the language of Hilbert series (or Poincar\'e series, or formal characters)
of modules over tca's.
