Given a group $G$ acting on a set $D$ of $n$ positions, and a set $R$ of colors (range) with a weight function $w: R \to \mathbb{Z}_{\geq 0}$, the figure counting series enumerates distinct colorings by weight. If the generating function for the colors is $f(x) = \sum_{r \in R} x^{w(r)}$, then the Pólya Enumeration Theorem expresses the generating function for distinct colorings.

The inventory (or pattern inventory) of colorings under a group action is obtained by substituting the figure counting series into the cycle index. If the "figure series" is $a + b + c + \cdots$ (one term per color), the inventory is: $Z_G(a + b + c + \cdots, a^2 + b^2 + c^2 + \cdots, \ldots, a^n + b^n + c^n + \cdots).$

The dihedral group $D_n$ of order $2n$ acts on $n$ positions by rotations and reflections. Its cycle index is: $Z_{D_n} = \frac{1}{2}Z_{C_n} + \begin{cases} \frac{1}{2}x_1 x_2^{(n-1)/2} & \text{if } n \text{ is odd}, \\ \frac{1}{4}\left(x_1^2 x_2^{(n-2)/2} + x_2^{n/2}\right) & \text{if } n \text{ is even}. \end{cases}$