The maximum number of Hamiltonian cycles in tournaments

Let $C(n)$ denote the maximum number of directed Hamiltonian cycles in a tournament on $n$ vertices. We prove that, as $n\to \mathrm{\infty }$, $$C(n)=(\frac{\cosh 1}{\cos 1}\pm o(1))\frac{(n-1)!}{2^n}\mathrm{.}$$ The bound is asymptotically attained by the regular tournament with vertices $\mathbb{Z}\mathrm{/}n\mathbb{Z}$ where $(i,j)$ is a directed edge if $j-i(\mathrm{m}\mathrm{o}\mathrm{d}n)<n\mathrm{/}2$. This answers a question of Alon and proves a conjecture of Wormald. Among the other implications of these results is asymptotically determining the maximum number of Hamiltonian paths in a tournament on $n$ vertices. We provide our own exposition of these proofs, which were originally found by GPT-5.6.