In "[The maximum number of Hamiltonian paths in tournaments](https://doi.org/10.1007/BF02128667)" by Noga Alon, the author states the following without proof (equation 3.1):

"Consider a random permutation $\pi$ of $\mathbb{Z}_n$. What is the probability that $\pi(i+1)−\pi(i) \mod{n} < n/2$ for all $i$?"

The claim is that this is $(2+o(1))^{−n}$, which makes sense and seems like it should be a standard argument. However, I have not been able to come up with a short proof, nor have I been able to find a proof in the literature.

Does anyone know of a complete proof?