What happens when you multiply a cycle index notation? I recently was introduced to this lovely combinatorial object https://en.wikipedia.org/wiki/Cycle_index#Applications. Now I'm curious if there are any operations we can do with it? Like if we square it or multiply it with another does the outcome have any relationship to the original two? Can we take derivatives? Can we take sums? Or is this just purely an organizational object with no capicity to be manipulated algebraically?
 A: The cycle index polynomial is an object attached to a permutation group, meaning an embedding $G \hookrightarrow S_n$ of a group into a symmetric group. Given two such permutation groups $G \hookrightarrow S_n, H \hookrightarrow S_m$, the product of their cycle index polynomials $Z(G) Z(H)$ is the cycle index polynomial $Z(G \times H)$ for the permutation group given by the embedding
$$G \times H \hookrightarrow S_n \times S_m \hookrightarrow S_{n+m}.$$
I'm not aware of any corresponding interpretation for sums. Derivatives correspond to generating functions for the number of cycles of a given length, and evaluating them with all variables set to $1$ tells you the average / expected number of such cycles. For example, using the exponential formula
$$\sum_{n \ge 0} Z(S_n) t^n = \exp \left( \sum_{k \ge 1} z_k \frac{t^k}{k} \right)$$
and taking derivatives with respect to each cycle variable $z_k$ you can compute (and there are other ways to do this computation) that the expected number of $k$-cycles in a random permutation in $S_n$ is $\frac{1}{k}$ (for $n \ge k$). By taking further derivatives you can compute higher moments also, and eventually learn that the number of $k$-cycles is asymptotically (as $n \to \infty$) Poisson distributed with rate $\lambda = \frac{1}{k}$. See this blog post for the calculation.
