Formulae and growth rates of high-dimensional polycubes

Abstract

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 2edo(d). We conjecture that it is asymptotically equal to (2d−3)e+O(1/d).

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Correspondence to Ronnie Barequet.

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Barequet, R., Barequet, G. & Rote, G. Formulae and growth rates of high-dimensional polycubes. Combinatorica 30, 257–275 (2010). https://doi.org/10.1007/s00493-010-2448-8

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Mathematics Subject Classification (2000)

  • 05A16
  • 05B50