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

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.