Irreducible characters of the symmetric group and exponential growth
  We consider sequences of degrees of ordinary irreducible $S_n$-characters. We assume that the corresponding Young diagrams have rows and columns bounded by some linear function of $n$ with leading coefficient less than one. We show that any such sequence has at least exponential growth and we compute an explicit bound. 
Antonio Giambruno and Sergey Mishchenko
math.CO math.RT