  Let $s_d(n)$ be the number of distinct decompositions of the $d$-dimensional
hypercube with $n$ rectangular regions that can be obtained via a sequence of
splitting operations. We prove that the generating series $y = \sum_{n \geq 1}
s_d(n)x^n$ satisfies the functional equation $x = \sum_{n\geq 1} \mu_d(n)y^n$,
where $\mu_d(n)$ is the $d$-fold Dirichlet convolution of the M\"obius
function. This generalizes a recent result by Goulden et al., and shows that
$s_1(n)$ also gives the number of natural exact covering systems of $\mZ$ with
$n$ residual classes. We also prove an asymptotic formula for $s_d(n)$ and
describe a bijection between $1$-dimensional decompositions and natural exact
covering systems.
