word representations of m×n×p proper arrays开题资料.pdf

: .
D author’s work on the enumeration of three-dimensional proper arrays [6]. The
building block of a proper array is a directed cube, where a directed cube is a cube which has five faces containing
circular indentations while the sixth face has a cylindrical plug or connector. See Fig. 1.
Fig. 1. A directed cube. In this illustration, the connector extends from the right face while the other five faces contain circular indentations.
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doi:. J. Quaintance / Discrete Mathematics 309 (2009) 1199–1212
Three-dimensional arrays can be built by fitting the connector of one directed cube into an indentation of another.
An m × n × p array of directed cubes is composed of p layers, with m rows and n columns in each layer. If the
m × n × p array of directed cubes obeys certain connectivity constraints, it is said to be an m × n × p proper array
[6]. An m × n × p proper array may be classified by its rightmost m × n layer of directed cubes, which is referred to
as the preferred face of the m × n × p proper array [6]. The preferred face of the m × n × p proper array represents
all its path-connected subsets and contains all its outward extending connectors.
The preferred face does not uniquely determine the number m × n × p proper arrays, since two different m × n × p
proper arrays may have the same preferred face. However, for fixed m and n and arbitrary p, the preferred face is the
state of a transition matrix Mm×n×p [1,2,12]. In particular, Mm×n×p is an r ×r matrix that describes how an m ×n × p
proper array evolves into an m × n × (p + 1) proper array. The number r is the number of possible preferred faces,
modulo symmetry, associated with m × n × p proper arrays