Number of permutations $\sigma\in S_n$ with $\sigma(k)\neq k$ for all $k=1,\ldots,n$

Question

For the symmetry group $S_n$ ($n\geq1$), how many permutations $\sigma$ exist with the property that $\sigma$ doesn't map any element of $\{1,\ldots,n\}$ to itself? I know I can try to do a counting argument for small $n$ (the first numbers of $n$ ($1,2,3,4$), calculated by hand, are $0,1,2,9$) but I would like to find a general method.