Permutations without two-cycles ($f(n)=n!{\sum }_{r=0}^{\lfloor n/2\rfloor }\frac{(-1{)}^r}{2^{r}r!}$)

These permutations have no cycle of length two, while fixed points and longer cycles are allowed. Inclusion-exclusion over specified two-cycles gives the count shown: only disjoint specified pairs can coexist. Their fraction among all permutations tends to $e^{-1/2}$. This condition differs from being a derangement and from refusing to express a permutation as a product of transpositions.