Counting tight Hamilton cycles in Dirac hypergraphs

Suppose $G$ is a $k$-uniform hypergraph on $n$ vertices such that every $(k-1)$-subset $S$ of $V(G)$ belongs to at least $\delta n$ edges, where $\delta >1\mathrm{/}2$. Let $\mathrm{\Psi }(G)$ denote the number of tight Hamilton cycles in $G$, that is, cyclic orderings of $V(G)$ in which every $k$ consecutive vertices form an edge. We prove that $\log \mathrm{\Psi }(G)\ge kh(G)-n\log \left(\genfrac{}{}{0px}{}{n}{k-1}\right)+n\log n-n\log e-o(n)$, where $h(G)$ is the hypergraph entropy of $G$, defined via perfect fractional matchings. This bound is tight, for example, for all (nearly) regular hypergraphs, in particular for the binomial random hypergraph. It also implies a conjecture by Ferber, Hardiman and Mond, stating that $\mathrm{\Psi }(G)\ge (\delta -o(1){)}^{n}n!$.