Derangements

Sets. Let $A_i=\{\sigma:\sigma(i)=i\}$. Count $\overline{A_1\cup\cdots\cup A_n}$.

Intersection size. $|A_{i_1}\cap\cdots\cap A_{i_k}|=(n-k)!$.

IE.
$$D_n=\sum_{k=0}^n(-1)^k\binom{n}{k}(n-k)!=n!\sum_{k=0}^n\frac{(-1)^k}{k!}.$$

Conclusion. $D_n=n!\sum_{k=0}^n\frac{(-1)^k}{k!}$.

Values. $D_3=2$, $D_4=9$, $D_5=44$.