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. Lattice animals, polyominoes, directed trees.

• Type: Journal
• Year: 2011
• Where: COCOON
• Authors: Andrei Asinowski, Gill Barequet, Ronnie Barequet, Günter Rote
• Keywords: COCOON 2011, Combinatorics, Discrete Geometry, Lattice Animals, Statistical Physics