The conjugacy classes of \(S_n\) are indexed by the partitions of \(n\) through cycle type, and so are its irreducible complex representations, the Specht modules \(S^\lambda\). The dimension of \(S^\lambda\) is the number of standard Young tableaux of shape \(\lambda\), and the Murnaghan–Nakayama rule evaluates the character \(\chi^\lambda\) on a class as a signed sum over successive removals of rim hooks. The Frobenius characteristic map is an isometry from the class functions on \(S_n\), with the character inner product, to the homogeneous symmetric functions of degree \(n\), with the Hall inner product; it sends \(\chi^\lambda\) to the Schur function \(s_\lambda\) and the character of \(\operatorname{Ind}_{S_m\times S_k}^{S_{m+k}}(V\boxtimes W)\) to the product of the images of \(V\) and \(W\). The Littlewood–Richardson coefficients \(c^\nu_{\lambda\mu}\) in \(s_\lambda s_\mu=\sum_\nu c^\nu_{\lambda\mu}s_\nu\) count Littlewood–Richardson tableaux. Pólya enumeration counts the orbits of a finite permutation group on colourings by Burnside’s lemma, through the cycle index of the group.