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In my earlier quest, we looked at $\chi_{\mu}^{\lambda}=$value of an irreducible character of the symmetric group $\frak{S}_n$, where $\mu$ and $\lambda$ are (unrestricted) partitions of $n$. Then, ...

Let $\chi_{\mu}^{\lambda}$ denote a value of an irreducible character of the symmetric group $\frak{S}_n$, where $\mu, \lambda\vdash n$. When $\mu=(n)$, then it's known that $$\sum_{\lambda\vdash n}\...