I am looking for the analytic formula for the following expression:
$$f_{n}(x,y)=\sum_{g\in S_{n}}x^{\chi(g)}y^{\chi(g\tau)}$$ where $g$ is taken from the permutation group $S_n$ the exponent () are number of cycles in the cycle notation of the cyclic permutation.
I have tried to use Mathematica to predict the general from the coefficients but no success. Probably the way to solve is using Polya's enumeration theorem, but I am not familiar with the topic.