The Perimeter of Proper Polycubes

We derive formulas for the number of polycubes of size $n$ and perimeter $t$ that are proper in $n-1$ and $n-2$ dimensions. These formulas complement computer based enumerations of perimeter polynomials in percolation problems. We demonstrate this by computing the perimeter polynomial for $n=12$ in arbitrary dimension $d$.Comment: 16 pages, 4 figures, 2 table

Proper n-Cell Polycubes in n − 3 Dimensions

A d-dimensional polycube of size n is a connected set of n cubes in d dimensions, where connectivity is through (d − 1)-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

Proper n-Cell Polycubes in n − 3 Dimensions

A d-dimensional polycube of size n is a connected set of n cubes in d dimensions, where connectivity is through (d−1)-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