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Combinatorica

, Volume 30, Issue 3, pp 257–275 | Cite as

Formulae and growth rates of high-dimensional polycubes

  • Ronnie Barequet
  • Gill Barequet
  • Günter Rote
Article

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).

Mathematics Subject Classification (2000)

05A16 05B50 

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Copyright information

© János Bolyai Mathematical Society and Springer Verlag 2010

Authors and Affiliations

  1. 1.Dept. of Mathematics and Dept. of Computer ScienceTel Aviv UniversityTel AvivIsrael
  2. 2.Dept. of Computer ScienceTechnion — Israel Institute of TechnologyHaifaIsrael
  3. 3.Institut für InformatikFreie Universität BerlinBerlinGermany

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