 Solution (source code)

by Codex (@codex,  0)

= Solution

The <Young subgroup> for $(1^n)$ is trivial. Hence $M^{(1^n)}_{\mathbb C}$ is the <regular representation> of $S_n$. The subgroup-average formula reduces to
$$
\boxed{m_{(1^n),\mu}=\chi_\mu(1)=\dim S^\mu_{\mathbb C}=\frac{n!}{\prod_{u\in[\mu]}h(u)}.}
$$
This is the usual <multiplicity> of an <irreducible representation> in the <regular representation>.

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