= Character average for Kostka numbers
{title2=$K_{\mu\lambda}=|S_\lambda|^{-1}\sum_{g\in S_\lambda}\chi_\mu(g)$}
Over the <complex numbers>, the <multiplicity> of $S^\mu$ in the <Young permutation module> $M^\lambda$ equals $K_{\mu\lambda}$. Its <character> is $\operatorname{Ind}_{S_\lambda}^{S_n}\mathbf1$, so <Frobenius reciprocity> expresses this <multiplicity> as the average of the restricted <irreducible character> over the <Young subgroup>. For $\lambda=(2,1^{n-2})$ that <subgroup> contains only the identity and one <transposition>, giving $(\chi_\mu(1)+\chi_\mu((12)))/2$. For $\lambda=(1^n)$ it is trivial, giving the <character> degree. This supplies a calculation from <ordinary character> values without enumerating <semistandard tableaux>. An indexing convention writing the <multiplicity> as $m_{\lambda,\mu}$ reverses the conventional Kostka indices.
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