If $\lambda = (\lambda_1,\lambda_2,\ldots)$ is a partition of $n$, then there is a permutation character of $S_n$ associated to the Young subgroup $S_\lambda$: $$ \pi_\lambda = \mathrm{Ind}_{S_\lambda}^{S_n}(1). $$ For a permutation $\sigma$, you can check that the formula for the value of $\pi_\lambda$ is given by $$ \pi_{\lambda} (\tau) = \frac{|\{ \sigma\in S_n : \sigma^{-1}\tau \sigma \in S_\lambda \}|}{|S_\lambda|}. $$
Is there an easy way to compute the number $|\{\sigma\in S_n : \sigma^{-1}\tau \sigma \in S_\lambda\}|$ given the cycle type of $\tau$?
If $\lambda = (\lambda_1,\lambda_2,\ldots,\lambda_l)$ is a partition of $n$ and $\tau\in S_n$ is a permutation of cycle type $\mu = (\mu_1,\mu_2,\ldots,\mu_r)$, then the value of the permutation character $\pi_\lambda(\tau)$ is given by the coefficient of $x_1^{\lambda_1}\cdots x_l^{\lambda_l}$ in the product $$ \prod_{i=1}^r (x_1^{\mu_i} + \cdots + x_l^{\mu_i}). $$
If you know of other ways to compute the value, please do still post your method.