For 0 ≤ \ell < k, a Hamiltonian \ell-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 \ell vertices.
We show that for all 0 ≤ r < k-1, every Dirac k-graph, that is, a k-graph with minimum co-degree pn for some p>1/2, has (up to a subexponential factor) at least as many Hamiltonian \ell-cycles as a typical random k-graph with edge-probability p.
This improves a recent result of Glock, Gould, Joos, Osthus and Kühn, and verifies a conjecture of Ferber, Krivelevich and Sudakov for all values 0 ≤ r < k-1.
