Ronnie Barequet, Gill Barequet, and Günter Rote:
Formulae and growth rates of high-dimensional polycubes
- In: European Conference on Combinatorics, Graph Theory and Applications
(EuroComb 2009), Bordeaux, September 2009. Editors: Jaroslav Nešetřil
and André Raspaud. Electronic
Notes in Discrete Mathematics 34 (2009), 459–463.
Combinatorica 30 (2010), 257–275. doi:10.1007/s00493-010-2448-8
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).
Last update: August 15, 2017.