Let C be a cyclic subgroup of S_n. How does the representation $Ind_C^{S_n}\rho$, where $\rho$ is some representation of $C$, decompose into irreducible components? Is there are a way to know which components appear with multiplicity 1?

There is a combinatorial way to decompose $Res_C^{S_n}S^\lambda$ for an irreducible $S_n$module $S^\lambda$. We use the notion of the "major index" for a standard tabuleau of shape $\lambda$. If $C=C_n$, the result is obtained by KraśkiewizWeyman. See also Garsia's paper "Combinatorics of the free Lie algebra and the symmetric group"(Theorem 8.4) and Reutenauer's book "Free Lie algebras"(Theorem 8.8 and 8.9). This result can be generalized to any cyclic subgroup in JöllenbeckSchocker's paper "Cyclic characters of symmetric groups". 

