Molien series

Definition
Let $$G$$ be a group and $$\rho:G \to GL(V)$$ be a finite-dimensional linear representation of $$G$$. The Molien series of $$\rho$$ is the formal power series:

$$\sum n_d t^d$$

where $$n_d$$ is the dimension of the subspace of $$Sym^d(V)$$ comprising those elements that are invariant under the action of $$G$$.

Equivalently, the map $$\rho:G \to GL(V)$$ gives an action of $$G$$ on the polynomial ring in $$dim(V)$$ variables. The dimension of the space of homogeneous degree $$d$$ polynomials that are invariant under this action, is the coefficient $$n_d$$ of $$t^d$$.

Facts
For a finite group, the Molien series corresponding to any finite-dimensional representation is a rational function of the formal variable $$t$$.