Enumeration of Specific Classes of Polycubes

Abstract

The aim of this paper is to gather several results concerning the enumeration of specific classes of polycubes. We first consider two classes of $3$-dimensional vertically-convex directed polycubes: the plateau polycubes and the parallelogram polycubes. An expression of the generating function is provided for the former class, as well as an asymptotic result for the number of polycubes of each class with respect to volume and width. We also consider three classes of $d$-dimensional polycubes $(d\geq 3)$ and we state asymptotic results for the number of polycubes of each class with respect to volume and width.

Enumeration of parallelogram polycubes

“…Morevover by summing for all possible heights of each plateau, we get the formula. The values of the diagonals corresponds to the results found experimentally in [5].…”

“…Unlike polyominoes, only a few classes of polycubes have been enumerated. Let us cite for instance, the plane partitions [7], the directed plateau polycubes [5], and the partially directed snake polycubes [11]. Some tools were developed for the enumeration of polycubes, in particular the generic method [6], an extension of Bousquet-Melou method [3] and the Dirichlet convolution for the enumeration of polycubes [4].…”

“…The class of parallelogram polycubes has been investigated. In [5], the authors found a differential equation for the generating function according the volume, width, the area of the rightmost face, the height of the last plateau and the depth of the last plateau, but this equation could not be solved. However, some asymptotic results, that are the only known, are given.…”

Directed Plateau Polyhypercubes

“…Polyhypercube are the extension of polyominoes in a dimension d ≥ 3 [1]. In Z d , a polyhypercube of dimension d is a finite union of cells (unit hypercubes), connected by their hypercubes of dimension d − 1, and defined up to translation [7]. Polyhypercubes are also called d-polycubes.…”

Plateau Polycubes and Lateral Area

“…A few classes of polycubes were studied. Among its, let us cite, for instance, the plane partitions [10], the directed plateau polycubes and also some asymptotic results concerning the parallelograms polycubes that were made according to the number of cells [7]. There exist two methods to enumerate families of polycubes.…”