Proper n-cell polycubes in n - 3 dimensions
COCOON'11 Proceedings of the 17th annual international conference on Computing and combinatorics
The growth rate of high-dimensional tree polycubes
European Journal of Combinatorics
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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).