= Permutations without two-cycles
{title2=$f(n)=n!\sum_{r=0}^{\lfloor n/2\rfloor}\frac{(-1)^r}{2^rr!}$}
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>.
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