Restricting representations of the symmetric group to the cyclic group

Restrict an irreducible representation $chi$ of $S_n$ over $mathbb C$ to a representation of the cyclic group of order $n$, $C_n$. Is there a combinatorial expression for how $chi$ restricted to $C_n$ splits into irreducible representations of $C_n$?

I am able to compute this using the inner product on characters of $C_n$, but I wonder if there is another description using Young diagrams/partitions of $n$.