The answer to a particular calculation in quantum information theory gives me the following expression:

Given $M$ specific elements of the symmetric group $S_n$, define the polynomial

$$Z_n(\pi_1, \dots , \pi_M ; X_i) = \sum_{\pi \in S_n} X_1^{C(\pi \cdot \pi_1)} X_2^{C(\pi \cdot \pi_2)} \cdots X_M^{C(\pi \cdot \pi_M)}$$

where $C(\pi)$ counts the number of disjoint cycles in the permutation $\pi$.

For $M=1$, this is related to the ordinary cycle index $Z(S_n; a_1, a_2, \dots)$ by

$$Z_n(\pi_1;X) = n! Z(S_n ; X,X, \dots)$$

My question is whether there is some useful discussion of this object $Z_n$ in the literature, or whether anyone can recommend an approach for calculating $Z_n$ for specific elements $(\pi_1,\dots, \pi_M)$ (a simple example of interest would be $\pi_1$ equal to the identity and $\pi_2$ equal to a cyclic permutation).