Formulae and growth rates of high-dimensional polycubes

A d-dimensional polycube is a facet-connected set of cubes in d dimensions. Fixed polycubes are considered distinct if they differ in their shape or orientation. A proper d-dimensional polycube spans all the d dimensions, that is, the convex hull of the centers of its cubes is d-dimensional. In this paper we prove rigorously some (previously conjectured) closed formulae for fixed (proper and improper) polycubes, and show that the growth-rate limit of the number of polycubes in d dimensions is 2ed - o(d). We conjecture that it is asymptotically equal to (2d - 3)e + O(1/d).

• Type: Journal
• Year: 2010
• Where: COMBINATORICA
• Authors: Ronnie Barequet, Gill Barequet, Günter Rote
• Keywords: COMBINATORICA 2010, D-dimensional Polycube, Dimensions, Machine Learning, Proper D-dimensional Polycube