Andrei Asinowski, Ronnie Barequet, Gill Barequet, and Günter Rote:
Proper n-cell polycubes in n−3 dimensions
In: "Computing and Combinatorics". Proceedings of the 17th
Annual International Computing and Combinatorics Conference
(COCOON 2011), Dallas, Texas, August 2011. Editors: Bin Fu
and Ding-Zhu Du. Lecture Notes in Computer Science 6842,
Springer-Verlag, 2011, pp. 181–191. doi:10.1007/978-3-642-22685-4_16
Journal of Integer Sequences
Article 12.8.4, 16 pages.
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.
Last update: October 3, 2012.