# Reference request: enumeration under group action

Is there a reference for the following lemma (which is useful in counting unlabeled k-trees)? It seems to me that it should be known, but I haven't been able to find it anywhere.

Let $G$ be a finite group acting on a weighted set $S$. We assume that each weight is a product of powers of variables. We do not assume that $S$ is finite, but we do assume that the sum of the weights of the elements of $S$ converges as a formal power series.

Let $M(S)$ be the set of multisets of elements of $S$. We may extend the action of $G$ on $S$ in an obvious way to an action on $M(S)$. We define the weight of a multiset in $M(S)$ to be the product of the weights of its elements. For any $g\in G$, we denote by $\operatorname{fix}(g)$ the sum of the weights of the elements of $S$ fixed by $G$. For any formal power series $u$ in the variables that appear in the weights, we define $p_n[u]$ to be the result of replacing each variable in $u$ by its $n$th power.

Let $g$ be an element of $G$. Then the sum of the weights of the elements of $M(S)$ fixed by $g$ is $$\exp\biggl(\sum_{m=1}^\infty \frac{p_m[\operatorname{fix}(g^{m})]}{m} \biggr).$$

This lemma is easy to prove by reducing to the case in which $g$ acts transitively on $S$, so I'm not looking for a proof, just a reference. (The case in which $g$ is the identity is well known.)

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