Decompositions of Unit Hypercubes and the Reversion of a Generalized M\"obius Series

TL;DR

This paper studies the enumeration of hypercube decompositions into rectangular regions, deriving a functional equation involving a generalized M"obius series, and provides asymptotic formulas and combinatorial bijections.

Contribution

It generalizes previous results by establishing a new functional equation for the generating series of hypercube decompositions using a generalized M"obius convolution.

Findings

Derived a functional equation relating the generating series and Dirichlet convolution.

Proved an asymptotic formula for the number of decompositions.

Established a bijection between 1D decompositions and exact covering systems.

Abstract

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.