Proper n-Cell Polycubes in n - 3 Dimensions

摘要

A d-dimensional polycube of size n is a connected set of n cubes in d dimensions, where connectivity is through (d- l)-dimensional faces. Enumeration of polycubes, and, in particular, specific types of polycubes, as well as computing the asymptotic growth rate of polycubes, is a popular problem in discrete geometry. This is also an important tool in statistical physics for computations related to percolation processes and branched polymers. In this paper we consider proper polycubes: A polycube is said to be proper in d dimensions if the convex hull of the centers of its cubes is d-dimensional. We prove a formula for the number of polycubes of size n that are proper in (n - 3) dimensions.