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In an answer to an earlier question, I showed how to prove that the square root counting function $r_2: S_n\rightarrow \mathbb{N},\;g\mapsto \#\{h\in S_n|h^2=g\}$ assumes its maximum at the identity, using the representation theory of $S_n$. Admittedly, you need to know slightly more than the character table. You need to be able to compute the Frobenius-Schur indicators of the characters, so you need to know how the conjugacy classes multiply. Alternatively, you just need to know that all representations are defined over $\mathbb{R}$, which you prove along to way to computing the character table anyway. In a comment to my answer, Richard Stanley remarks that, also using the representation theory of $S_n$, you can generalise this to the $k$-th root counting function for any positive integer $k$. In an answer to the same question, Alon Amit remarks on possible generalisations to solving other polynomial equations in the elements of $S_n$.