Creating single-cycle permutations
The obvious way to construct such a pseudorandom single-cycle permutation is to take a pseudorandom permutation $P$ (which need not be single-cycle), a simple fixed single-cycle permutation $Inc$ (e.g. just increment the value by 1), and construct:

$$S = P^{-1} \ \circ Inc \ \circ P $$

That is, to evaluate $S(x)$, you first apply the permutation $P$ to $x$, then the permutation $Inc$, and then the permutation $P^{-1}$, which is the inverse of the permutation $P$.

Things that should be obvious:


$S$ is a single cycle permutation
Every single cycle permutation can be expressed this way with an appropriate choice of $P$
(Less obvious) if $P$ is selected uniformly over all permutations, then $S$ is selected uniformly over all single-cycle permutations.