Polya's theory of counting and commutative algebra
Pólya's theorem arises in the theory of invariants of finite groups. See Section 10 of http://math.mit.edu/~rstan/pubs/pubfiles/38.pdf.
I must admit that I don't understand what the posted question is asking, but since the OP seemed to be satisfied with Professor Stanley's answer, I would like to add an algebro-geometric analogue of this situation that may be relevant to this, at least in spirit. (Thanks for asking this question and the answers, which gave me more things to think about, especially from Stanley's article.)
When $X$ is a finite set, studying the enumeration in the set-theoretic quotient $X^{n}/G$ is the main purpose of the Pólya enumeration theorem, and studying the quotient space $X^{n}/G$, when $X$ is a finite CW complex, can be thought as a topological analogue. What can we count about this quotient space? Among various options, we can count "holes", or more precisely, the singular Betti numbers $h^{i}(X^{n}/G)$. Writing
$$\chi_{u}(Y) := \sum_{i=0}^{\infty}(-u)^{i}h^{i}(Y),$$
Macdonald showed that
$$\chi_{u}(X^{n}/G) = Z_{G}(\chi_{u}(X), \chi_{u^{2}}(X), \dots, \chi_{u^{n}}(X)),$$
where
$$Z_{G}(x_{1}, \dots, x_{n}) := \frac{1}{|G|}\sum_{g \in G}x_{1}^{m_{1}(g)} \cdots x_{n}^{m_{n}(g)},$$
writing $m_{i}(g)$ to mean the number of length $i$ cycles in the cycle decomposition of $g$ in $S_{n}$, which is the cycle index of $G$ in $S_{n}$ appearing in the original Pólya enumeration theorem.
My favorite analogue is to consider this over a finite field $\mathbb{F}_{q}$. Namely, if $X$ is a quasi-projective variety over $\mathbb{F}_{q}$, then it turns out that
$$|(X^{n}/G)(\mathbb{F}_{q})| = Z_{G}(|X(\mathbb{F}_{q})|, |X(\mathbb{F}_{q^{2}})|, \dots, |X(\mathbb{F}_{q^{n}})|),$$
where $G$ acts on $X^{n}$ by permuting coordinates (as in the set-theoretic setting). When $X = \mathbb{A}^{1}_{\mathbb{F}_{q}} = \mathrm{Spec}(\mathbb{F}_{q}[t])$, this statement says that there are precisely $q^{n}$ algebra maps
$$\mathbb{F}_{q}[x_{1}, \dots, x_{n}]^{G} \rightarrow \mathbb{F}_{q}$$
over $\mathbb{F}_{q}$, where $G$ acts on the polynomial algebra by permuting variables.
One place you can find these sorts of analogues is my recent preprint: Pólya enumeration theorems in algebraic geometry.