In Richard Stanley's Enumerative Combinatorics, Vol. 2, the following is an exercise:
Let $k\in\mathbb{N}$. Show that $\sum_{w\in S_n}{p_{\rho(w^k)}}$ is a nonnegative integer linear combination of Schur functions. Equivalently, the function $r_k=r_{n,k} : S_n \rightarrow \mathbb{Z}$ defined by $r_k(w)=~\#\{u\in S_n : u^k = w\}$ is a character of $S_n$.
I do not understand how these statements are equivalent.