A Note on the Symmetric Powers of the Standard Representation of S_n 
David Savitt, Richard P. Stanley 

Abstract: 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. 

Math Subject Class: 05E10