  In this paper, we prove that the dimension of the space spanned by the
characters of the symmetric powers of the standard n-dimensional representation
of the symmetric group S_n is asymptotic to n^2/2. This is proved by using
generating functions to obtain formulas for upper and lower bounds, both
asymptotic to n^2/2, for this dimension. In particular, for n>6, these
characters do not span the full space of class functions on S_n.
