Let $Q_n$ be the $n$-dimensional [hypercube graph](https://en.wikipedia.org/wiki/Hypercube_graph).  How many [vertex cycle covers](https://en.wikipedia.org/wiki/Vertex_cycle_cover) exist on $Q_n$?  (Presumably the best we can hope for are upper and lower bounds.)  To be clear, a single "vertex cycle cover" is a set of cycles in $Q_n$ such that each vertex is a member of one and only one cycle.

To fix some notation, like $N=2^n$ and let $C_n$ be the count of the number of vertex cycle covers on $Q_n$.

Note that there are about $(n/e)^N$ Hamiltonian cycles in $Q_n$ (cf [Feder and Subi, 2008](theory.stanford.edu/~tomas/hamat.ps) for more precise upper and lower bounds), which provides a lower bound on $C_n$.