Symmetric group character degrees and hook numbers

In this article we prove the following result: for any two naturalnumbers k and , and for all sufficiently large symmetric groupsS_n, there are k disjoint sets of irreducible characters ofS_n, such that each set consists of characters with the samedegree, and distinct sets have different degrees. In particular,this resolves a conjecture most recently made by Moretóin [5]. The methods employed here are based upon the dualitybetween irreducible characters of the symmetric groups and thepartitions to which they correspond. Consequently, the paperis combinatorial in nature.