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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.

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    Thank you! The link is very helpful. If $w\in S_n$, then the \textit{cycle type} $\rho(w)$ is the partition $\rho(w)=(\rho_1, \rho_2, ...) \vdash n$ such that the cycle lengths of w (in its factorization into disjoint cycles) are $\rho_1, \rho_2, \ldots$.2011-04-24

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