The Perimeter of Proper Polycubes

TL;DR

This paper derives formulas for counting proper polycubes with given size and perimeter in various dimensions, complementing computational enumeration methods and providing explicit perimeter polynomials for specific cases.

Contribution

It introduces new formulas for enumerating proper polycubes of size n and perimeter t in different dimensions, enhancing theoretical understanding and computational approaches.

Findings

Formulas for polycube counts with specific perimeters in various dimensions

Perimeter polynomial computed for n=12 in arbitrary dimension d

Complemented existing computer-based enumeration methods

Abstract

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 $d$.