Counting n-cell polycubes proper in n−k 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. In this paper, we develop a theoretical framework for computing the explicit formula enumerating polycubes of size n that span n−k dimensions, for a fixed value of k. Besides the interest in the number of these simple combinatorial objects, known as proper polycubes, such formulae play an important role in the literature of statistical physics in the study of percolation processes and collapse of branched polymers. The main contribution of this framework is that it enabled us to prove a conjecture about the general form of the formula for a general k. We also used this framework for implementing a computer program which reaffirmed the known formulae for k=2 and k=3, and proved rigorously, for the first time, the formulae for k=4 and k=5.