Example 4.4.2 Color the 3 vertices of an equilateral triangle with red, blue, and green. How many different schemes are there? If (1) schemes that can be superimposed by rotation are considered the same, (2) schemes that can be superimposed by rotation and reflection are considered the same.


Solution 1: $D=\{1,2,3\}, R=\{r, b, g\}, C$ has a total of $3^{3}=27$ coloring methods. Using Burnside's formula to calculate:
(1)
$$
\begin{array}{l}
G=\{e,(1,2,3),(1,3,2)\}, \\
N(G, C)=\frac{1}{3}(27+3+3)=11 .
\end{array}
$$
(2)
$$
\begin{array}{l}
G=\{e,(1,2,3),(1,3,2),(1,3),(1,2),(2,3)\}, \\
N(G, C)=\frac{1}{6}\left(27+3+3+3^{2} \times 3\right)=10 .
\end{array}
$$

Solution 2: (1) Using Pólya's formula:
$$
\begin{array}{l}
G=\{e,(1,2,3),(1,3,2)\}, P_{G}\left(x_{1}, x_{2}, x_{3}\right)=\frac{1}{3}\left(x_{1}^{3}+2 x_{3}\right), \\
P_{G}(3,3,3)=\frac{1}{3}\left(3^{3}+2 \times 3\right)=11 .
\end{array}
$$
$$
\begin{array}{l}
G=\{e,(1,2,3),(1,3,2),(1,3),(1,2),(2,3)\}, \\
P_{G}\left(x_{1}, x_{2}, x_{3}\right)=\frac{1}{6}\left(x_{1}^{3}+2 x_{3}+3 x_{1} x_{2}\right), \\
P_{G}(3,3,3)=\frac{1}{6}\left(3^{3}+2 \times 3+3 \times 3^{2}\right)=10 .
\end{array}
$$