Partitioning the hypercube into smaller hypercubes

TL;DR

This paper investigates how the vertices of a d-dimensional hypercube can be partitioned into smaller hypercubes, providing asymptotic estimates and exploring related combinatorial questions.

Contribution

It offers the first asymptotic bounds on the number of hypercube partitions into smaller hypercubes and connects this to perfect matchings in hypercubes.

Findings

Asymptotic order of partition count estimated

Relation established between partitions and perfect matchings

New questions posed in hypercube combinatorics

Abstract

Denote by Q_d the d-dimensional hypercube. Addressing a recent question we estimate the number of ways the vertex set of Q_d can be partitioned into vertex disjoint smaller cubes. Among other results, we prove that the asymptotic order of this function is not much larger than the number of perfect matchings of Q_d. We also describe several new (and old) questions.