2. Given $A=\{1,2,3,4,5,6\}, f: A \rightarrow A$, the number of mappings $f$ that satisfy $f(f(x)) \neq x$ is $\qquad$
2.7360 .

First, there should be $f(x) \neq x$, and the number of mappings satisfying $f(x) \neq x$ is $5^{6}$.
If $x \rightarrow a, a \rightarrow x(a \neq x)$, then it is called a pair of cyclic correspondences.
If there is one pair of cyclic correspondences and $f(x) \neq x$, the number of such mappings is $C_{6}^{2} \cdot 5^{4}$; if there are two pairs of cyclic correspondences and $f(x) \neq x$, the number of such mappings is $\frac{\mathrm{C}_{6}^{2} \mathrm{C}_{4}^{2}}{\mathrm{~A}_{2}^{2}} \cdot 5^{2}$; if there are three pairs of cyclic correspondences and $f(x) \neq x$, the number of such mappings is $\frac{C_{6}^{2} C_{4}^{2} C_{2}^{2}}{A_{3}^{3}}$.
Therefore, the number of mappings that meet the conditions is
$$
5^{6}-C_{6}^{2} \cdot 5^{4}+\frac{C_{6}^{2} C_{4}^{2}}{A_{2}^{2}} \cdot 5^{2}-\frac{C_{6}^{2} C_{4}^{2} C_{2}^{2}}{A_{3}^{3}}=7360
$$