For example, $1 f: A_{3} \rightarrow A_{3}$, find the number of $f$ that satisfy $f^{(3)}=f$, and list all such $f$.

 untranslated text remains as:
例 $1 f: A_{3} \rightarrow A_{3}$, 求满足 $f^{(3)}=f$ 的 $f$ 个数, 并作出所有这些 $f$. 

However, the requested translation is:
For example, $1 f: A_{3} \rightarrow A_{3}$, find the number of $f$ that satisfy $f^{(3)}=f$, and list all such $f$.
Solution: $n=3, m=3, k_{i} \mid 3-1, k_{i}$ can take 1 or 2.
$k=1, 1 \cdot x_{1}=1$ has solution $x_{1}=1$;
$k=2, 1 \cdot x_{1}=2$ has solution $x_{1}=2; 2 \cdot x_{2}=2$ has solution $x_{2}=1$;
$k=3, 1 \cdot x_{1}=3$ has solution $x_{1}=3; 1 \cdot x_{1}+2 \cdot x_{2}=3$ has solution $x_{1}=1, x_{2}=$
$A(3,1)=1$;
$$
\begin{aligned}
A(3,2) & =\frac{2!}{1^{2} \cdot 2!}+\frac{2!}{2^{1} \cdot 1!}=2 ; \\
A(3,3) & =\frac{3!}{1^{3} \cdot 3!}+\frac{3!}{1^{1} \cdot 2^{1} \cdot 1! \cdot 1!}=4 ; \\
B(3,3) & =1^{3-1} \mathrm{C}_{3}^{1} A(3,1)+2^{3-2} \mathrm{C}_{3}^{2} A(3,2)+3^{3-3} \mathrm{C}_{3}^{3} A(3,3) \\
& =1 \cdot 3 \cdot 1+2 \cdot 3 \cdot 2+1 \cdot 1 \cdot 4=19 .
\end{aligned}
$$

Construct these 19 $f$ as follows:
$$
\begin{array}{l}
k=1, \quad\left(\begin{array}{lll}
1, & 2, & 3 \\
1, & 1, & 1
\end{array}\right), \quad\left(\begin{array}{lll}
1, & 2, & 3 \\
2, & 2, & 2
\end{array}\right), \quad\left(\begin{array}{lll}
1, & 2, & 3 \\
3, & 3, & 3
\end{array}\right) . \\
k=2, \quad\left(\begin{array}{lll}
1, & 2, & 3 \\
1, & 2, & 1
\end{array}\right),\left(\begin{array}{lll}
1, & 2, & 3 \\
1, & 1, & 3
\end{array}\right),\left(\begin{array}{lll}
1, & 2, & 3 \\
2, & 2, & 3
\end{array}\right),\left(\begin{array}{lll}
1, & 2, & 3 \\
1, & 2, & 2
\end{array}\right), \\
\left(\begin{array}{lll}
1, & 2, & 3 \\
1, & 3, & 3
\end{array}\right), \quad\left(\begin{array}{lll}
1, & 2, & 3 \\
3, & 2, & 3
\end{array}\right),\left(\begin{array}{lll}
1, & 2, & 3 \\
2, & 1, & 1
\end{array}\right),\left(\begin{array}{lll}
1, & 2, & 3 \\
3, & 1, & 1
\end{array}\right), \\
\left(\begin{array}{lll}
1, & 2, & 3 \\
2, & 3, & 2
\end{array}\right),\left(\begin{array}{lll}
1, & 2, & 3 \\
2, & 1, & 2
\end{array}\right),\left(\begin{array}{lll}
1, & 2, & 3 \\
3, & 3, & 1
\end{array}\right),\left(\begin{array}{lll}
1, & 2, & 3 \\
3, & 3, & 2
\end{array}\right), \\
k=3, \quad\left(\begin{array}{lll}
1, & 2, & 3 \\
1, & 2, & 3
\end{array}\right) \cdot\left(\begin{array}{lll}
1, & 2, & 3 \\
2, & 1, & 3
\end{array}\right) \cdot\left(\begin{array}{lll}
1, & 2, & 3 \\
3, & 2, & 1
\end{array}\right),\left(\begin{array}{lll}
1, & 2, & 3 \\
1, & 3, & 2
\end{array}\right) \cdot \\
\end{array}
$$