Past exam of the representation Theory of Symmetric Groups course of the University of Oxford / 2011 / 1 / c / Solution

The Young subgroup for $(1^n)$ is trivial. Hence $M_{\mathbb{C}}^{(1^n)}$ is the regular representation of $S_n$. The subgroup-average formula reduces toThis is the usual multiplicity of an irreducible representation in the regular representation.