On the Number of Hamiltonian Cycles in a Boolean Cube

It is shown that, as $n\to \mathrm{\infty }$, the logarithm of the number of decompositions into cycles of the $n$-dimensional Boolean cube $E^n$ is $$2^n(\ln n-1+o(1)),$$ and the logarithm of the number of Hamiltonian cycles in $E^n$ is at least $$2^{n-1}(\ln n-1+o(1))\mathrm{.}$$ It is proved that, in $E^n$, every perfect matching whose edges belong to at most $k$ directions can be extended to a Hamiltonian cycle for every $n\ge n_0(k)$.