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≥ 2 and 1≤ ≤ k-1 , we show that every k -graph on n vertices with minimum codegree at least {equation*} \{ {array}{l@{}l} (1/2+o(1) )n & if (k- ) k,\\[5pt] ( {1}{ k/k- (k- )}+o(1) )n & if (k- ) k, {array} . {equation*} contains exp\!(nlog n-Θ (n)) Hamilton -cycles as long as (k- ) n . When (k- ) k , this gives a simple proof of a result of Glock, Gould, Joos, Kühn, and Osthus, while when (k- ) k , this gives a weaker count than that given by Ferber, Hardiman, and Mond, or when < k/2 , by Ferber, Krivelevich, and Sudakov, but one that holds for an asymptotically optimal minimum codegree bound.