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 = ₍ ₁ sd (n) xⁿ satisfies the functional equation x = ₍ ₁ d (n) yⁿ, where d (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.