Abstract
A d-dimensional polycube of size n is a connected set of n cubes in d dimensions, where connectivity is through (d − 1)-dimensional faces. Enumeration of polycubes, and, in particular, specific types of polycubes, as well as computing the asymptotic growth rate of polycubes, is a popular problem in discrete geometry. This is also an important tool in statistical physics for computations related to percolation processes and branched polymers. In this paper we consider proper polycubes: A polycube is said to be proper in d dimensions if the convex hull of the centers of its cubes is d-dimensional. We prove a formula for the number of polycubes of size n that are proper in (n − 3) dimensions.
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Asinowski, A., Barequet, G., Barequet, R., Rote, G. (2011). Proper n-Cell Polycubes in n − 3 Dimensions. In: Fu, B., Du, DZ. (eds) Computing and Combinatorics. COCOON 2011. Lecture Notes in Computer Science, vol 6842. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-22685-4_16
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DOI: https://doi.org/10.1007/978-3-642-22685-4_16
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-22684-7
Online ISBN: 978-3-642-22685-4
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