In http://www.springerlink.com/content/y19u81675243r237/fulltext.pdf, 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?

In "The maximum number of Hamiltonian paths in tournaments" 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?