Decompositions of Unit Hypercubes and the Reversion of a Generalized Möbius Series

Let sd(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 = ∑n ≥ 1 sd(n)xⁿ satisfies the functional equation x = ∑n≥ 1 μd(n)yⁿ, where μd(n) is the d-fold Dirichlet convolution of the Möbius function. This generalizes a recent result by Goulden et al., and shows that s₁(n) also gives the number of natural exact covering systems of Z with n residual classes. We also prove an asymptotic formula for sd(n) and describe a bijection between $1$-dimensional decompositions and natural exact covering systems.