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Formulae and growth rates of high-dimensional polycubes

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

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Authors and Affiliations

  1. Dept. of Mathematics and Dept. of Computer Science, Tel Aviv University, Tel Aviv, 69978, Israel

    Ronnie Barequet

  2. Dept. of Computer Science, Technion — Israel Institute of Technology, Haifa, 32000, Israel

    Gill Barequet

  3. Institut für Informatik, Freie Universität Berlin, Takustraße 9, D-14195, Berlin, Germany

    Günter Rote

Authors
  1. Ronnie Barequet
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  2. Gill Barequet
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  3. Günter Rote
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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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  • DOI: https://doi.org/10.1007/s00493-010-2448-8

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