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The sum $\sum_\chi \chi(\alpha)^2$ is the size of the centralizer $z_\mu=\frac{n!}{|K_\mu|}=1^{m_1}m_1! 2^{m_2}m_2!\cdots n^{m_n} m_n!$ if $\alpha$ has cycle type $\mu=1^{m_1}2^{m_2}\ldots$, so
$$\...

The sum $\sum_\mu \chi^\lambda_\mu$ over partitions $\mu$ of $n$ is the multiplicity of the irreducible $\chi^\lambda$ in the character afforded by $\mathfrak S_n$ acting on itself by conjugation. If $...

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