A specialization of the Polya enumeration theorem can give rise to the following identity:

$\frac{1}{k!} \sum_{\pi \in S_k} n^{cyc(\pi)} = {n+k-1 \choose k}, $

where $S_k$ is the permutation group of degree $k$ and $cyc(\pi)$ is the number of cycles in the permutation $\pi$. Now, I am interested in the following summation

$\sum_{\pi \in S_k} n_1^{cyc(\pi)} n_2^{cyc(\pi \xi )} n_3^{cyc(\pi \xi^{-1})}, $

where $\xi$ is the cyclic permutation of the cycle type $(123....k)$. **Is there an expression in closed form for this summation just like the previous summation? Is this summation related to the Polya enumeration theorem in some way？** We might need to separately discuss the two cases with even $k$ and odd $k$. The naive reasoning is that, for example, if we look at the term with the highest power in, say $n_2$, we get $n_1n_2^k n_3$ for odd $k$ and $n_1n_2^k n_3^2$ for even $k$.