 Character average for Kostka numbers

Over the complex numbers, the multiplicity of $S^{\mu }$ in the Young permutation module $M^{\lambda }$ equals $K_{\mu \lambda }$. Its character is ${\mathrm{Ind}}_{S_{\lambda }}^{S_n}1$, 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.