Example 7.3 Let $A=\{1,2,3\}$, find $P_{S_{4}}\left(x_{1}, x_{2}, x_{3}\right)$.
Solution: Since $S_{A}=\{(1)(2)(3),(1)(23),(2)(13),(3)(12),(123),(132)\}$, therefore $P_{S_{4}}\left(x_{1}, x_{2}, x_{3}\right)=\frac{1}{6}\left(x_{1}^{3}+3 x_{1} x_{2}+2 x_{3}\right)$.