Proper n-Cell Polycubes in n − 3 Dimensions

Abstract

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 combinatorics and discrete geometry. This is also an important tool in statistical physics for computations and analysis of percolation processes and collapse of branched polymers. A polycube is said to be proper in d dimensions if the convex hull of the centers of its cubes is d-dimensional. In this paper we prove that the number of polycubes of size n that are proper in n − 3 dimensions is 2 n−6 n n−7 (n − 3)(12n 5 − 104n 4 + 360n 3 − 679n 2 + 1122n − 1560)/3.

Automatic Proofs for Formulae Enumerating Proper Polycubes

“…The proof uses a case analysis of the possible structures of spanning trees of the polycubes, and the various ways in which cycles can be formed in their cell-adjacency graphs. Similarly, Asinowski et al [1] proved that DX(n, n − 3) = 2 n−6 n n−7 (n − 3)(12n 5 − 104n 4 + 360n 3 − 679n 2 + 1122n − 1560)/3, again, by counting spanning trees of polycubes, yet the reasoning and the calculations were significantly more involved. The inclusion-exclusion principle was applied in order to count correctly polycubes whose cell-adjacency graphs contained certain subgraphs, so-called "distinguished structures."…”

The Perimeter of Proper Polycubes

“…On the other hand some labeled spanning trees represent impossible polycubes with overlapping cells. Yet a careful consideration of all these cases has yield the formulas for DX(n, n − 2) [9] and DX(n, n − 3) [10]. Using non-rigorous arguments, DX(n, n − k) has been computed up to k = 7 [5].…”