Bounded-displacement permutation of the natural numbers
= Bounded-displacement permutation of the natural numbers
A permutation $\sigma$ of $\mathbb N$ has displacement at most one when $|\sigma(j)-j|\leq1$. Such a permutation is a disjoint collection of fixed points and adjacent transpositions, so independently swapping the pairs $(2n-1,2n)$ already gives uncountably many examples.