Andrei Asinowski, Ronnie Barequet, Gill Barequet, and Günter Rote:
Proper ncell 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 DingZhu Du. Lecture Notes in Computer Science 6842,
SpringerVerlag, 2011, pp. 181–191. doi:10.1007/9783642226854_16

Journal of Integer Sequences
15 (2012),
Article 12.8.4, 16 pages.
Abstract
A ddimensional 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 ddimensional. We
prove a formula for the number of polycubes of size n that are
proper in n−3 dimensions.
Last update: October 3, 2012.