Question 183, Let set $\mathrm{A}=\{1,2,3,4,5,6,7\}$, f: $\mathrm{A} \rightarrow \mathrm{A}$ is a bijection, and satisfies $\mathrm{f}(\mathrm{f}(\mathrm{f}(\mathrm{x})))=$ $\mathrm{x}, \mathrm{x} \in \mathrm{A}$, try to find the number of such mappings $\mathrm{f}$.
Question 183, Answer: Construct a directed graph $G(V, E)$, where the 7 points in $V$ represent the 7 elements in $A$. If $f(a)=b$, then connect a directed edge $\overline{a b}$. Since $f$ is a bijection, each point in $V$ has an outdegree and indegree of 1, thus $G$ can be divided into several disjoint directed cycles, and the lengths of these cycles are all divisors of 3, meaning each cycle's length is either 1 or 3. The number of graphs without cycles of length 3 is 1; the number of graphs with exactly one cycle of length 3 is $\frac{A_{7}^{3}}{3}=70$; the number of graphs with exactly two cycles of length 3 is $\frac{\frac{13}{3} A_{1}^{3}}{3}=280$. Therefore, the number of $\mathrm{f}$ that satisfy the condition is $1+70+280=351$.