Counting spanning subgraphs in dense hypergraphs

We give a simple method to estimate the number of distinct copies of some classes of spanning subgraphs in hypergraphs with a high minimum degree. In particular, for each $k\geq 2$ and $1\leq \ell \leq k-1$ , we show that every $k$ -graph on $n$ vertices with minimum codegree at least \begin{equation*} \left \{\begin {array}{l@{\quad}l} \left (\dfrac {1}{2}+o(1)\right )n & \text { if }(k-\ell )\mid k,\\[5pt] \left (\dfrac {1}{\lceil \frac {k}{k-\ell }\rceil (k-\ell )}+o(1)\right )n & \text { if }(k-\ell )\nmid k, \end {array} \right . \end{equation*} contains $\exp\!(n\log n-\Theta (n))$ Hamilton $\ell$ -cycles as long as $(k-\ell )\mid n$ . When $(k-\ell )\mid k$ , this gives a simple proof of a result of Glock, Gould, Joos, Kühn, and Osthus, while when $(k-\ell )\nmid k$ , this gives a weaker count than that given by Ferber, Hardiman, and Mond, or when $\ell \lt k/2$ , by Ferber, Krivelevich, and Sudakov, but one that holds for an asymptotically optimal minimum codegree bound.