Counting Hamiltonian Cycles in Dirac Hypergraphs

For 0 6 ` < k, a Hamiltonian `-cycle in a k-uniform hypergraph H is a cyclic ordering of the vertices of H in which the edges are segments of length k and every two consecutive edges overlap in exactly ` vertices. We show that for all 0 6 ` < k − 1, every k-graph with minimum co-degree δn with δ > 1/2 has (asymptotically and up to a subexponential factor) at least as many Hamiltonian `-cycles as in a typical random k-graph with edgeprobability δ. This significantly improves a recent result of Glock, Gould, Joos, Kühn, and Osthus, and verifies a conjecture of Ferber, Krivelevich and Sudakov for all values 0 6 ` < k − 1.