Formulae and growth rates of high-dimensional polycubes

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).

Proper n-Cell Polycubes in n − 3 Dimensions

“…This 1/d-expansion (of the free energy of animals, in their terminology) is partly based on so-called "diagonal formulae," that is, formulae for DX(n, n − k), where k > 0 is a small constant. It turned out that this expansion is consistent with the main result obtained by Barequet et al [2], namely, that the growth-rate limit of the number of polycubes in d dimensions is asymptotically 2ed − o(d), conjectured to asymptotically be (2d − 3)e − 31e 48d…”

“…In counting directed trees with n − 1 labeled edges, which have subgraphs as in Figure 2, two lemmas will be used. Lemma 4 was proved earlier [2]; we will here relate it to a result from the literature.…”

“…Thus, the distinguished structures are A, I, B, C, and other structures that contain several occurrences of these "basic" structures. 2 The number of occurrences is limited, since each occurrence uses up some repeated labels. The enumeration of the distinguished structures is, thus, a finite task.…”

“…Similarly to our proof [2] of the formula for DX(n, n − 2), we prove the formula for DX(n, n − 3) by counting spanning trees of polycubes, yet the reasoning and the calculations are significantly more involved. Indeed, we use the inclusion-exclusion principle in order to count correctly polycubes whose cell-adjacency graphs contain cycles.…”

“…Barequet et al [2] proved rigorously, for the first time, that DX(n, n − 2) = 2 n−3 n n−5 (n − 2)(2n 2 − 6n + 9)…”