2. Let the set $A=\{1,2,3,4,5,6\}$, and the mapping $f: A$
$\rightarrow A$ satisfies $f(f(x))=x$. Then the number of mappings $f$ is
$\qquad$ ـ.
2.76.

Let $f(x)=a$, then $f(a)=x$, i.e., $x \rightarrow a, a \rightarrow x$.
If $a=x$, i.e., $x \rightarrow x$ is called self-correspondence;
If $a \neq x$, i.e., $x \rightarrow a, a \rightarrow x$ is called a pair of cyclic correspondence.
It is also known that the number of self-correspondences should be even, so the total number of mappings is
$$
\begin{array}{l}
\mathrm{C}_{6}^{6}+\mathrm{C}_{6}^{4} \cdot 1+\mathrm{C}_{6}^{2} \cdot \frac{\mathrm{C}_{4}^{2} \mathrm{C}_{2}^{2}}{\mathrm{P}_{2}^{2}}+\mathrm{C}_{6}^{0} \cdot \frac{\mathrm{C}_{6}^{2} \mathrm{C}_{4}^{2} \mathrm{C}_{2}^{2}}{\mathrm{P}_{3}^{3}} \\
=1+15+45+15=76 .
\end{array}
$$