Example 7.9 Let $A_{n}$ be a regular $n$-sided polygon. Now, we use $m$ colors to paint the $n$ sides of $A_{n}$, with each side painted one color. Find the number of type I patterns of the edge-colored regular $n$-sided polygon.
Let $(C, *)$ be the normal rotation group of a regular $n$-sided polygon $A_{n}$, and the edge permutation group it induces be $\left(\boldsymbol{G},{ }^{\circ}\right)$. Suppose $c \in C$, and the edge permutation it induces is $g$. Let $A_{n}^{\prime}$ be the edge-colored regular $n$-sided polygon obtained by coloring the $n$ edges of $A_{n}$. Then $c A_{n}^{\prime}=A_{n}^{\prime}$ if and only if the edges of $A_{n}^{\prime}$ in the same cycle of $g$ have the same color. Since
$$
P_{G}\left(x_{1}, x_{2}, \cdots, x_{n}\right)=\frac{1}{n} \sum_{d \mid n} \varphi(d) \cdot\left(x_{d}\right)^{\frac{n}{d}},
$$

by Burnside's Lemma, the number of distinct edge-colored regular $n$-sided polygons is
$$
N=\frac{1}{n} \sum_{d \backslash n} \varphi(d) m^{\frac{n}{d}} .
$$