Counting Hamilton cycles in Dirac hypergraphs

A tight Hamilton cycle in a $k$-uniform hypergraph ($k$-graph) $G$ is a cyclic ordering of the vertices of $G$ such that every set of $k$ consecutive vertices in the ordering forms an edge. R{o}dl, Ruci{n}ski, and Szemer{e}di proved that for $kgeq 3$, every $k$-graph on $n$ vertices with minimum codegree at least $n/2+o(n)$ contains a tight Hamilton cycle. We show that the number of tight Hamilton cycles in such $k$-graphs is $exp(nln n-Theta(n))$. As a corollary, we obtain a similar estimate on the number of Hamilton $ell$-cycles in such $k$-graphs for all $ellin{0,dots,k-1}$, which makes progress on a question of Ferber, Krivelevich and Sudakov.