Word representations of m x n x p proper arrays

Let m not equal n. An m x n x p proper array is a three-dimensional array composed of directed Cubes that obeys certain constraints. Due to these constraints, the m x n x p proper arrays may be classified via a schema in which each m x n x p proper array is associated with a particular m x n planar face. By representing each connected component present in the m x n planar face with a distinct letter, and the position of each outward pointing connector by a circle, all in x it array of circled letters is formed. This m x n array of circled letters is the word representation associated with the m x n x p proper array. The main result of this paper provides an upper bound for the number of all m x n word representations modulo symmetry, where the symmetry is derived from the group D-2 = C-2 x C-2 acting on the set of word representations. This upper bound is achieved by forming a linear combination of four exponential generating functions, each of which is derived front a particular symmetry operation. This linear combination Counts the number of partitions of the set of m x it circled letter arrays that are inequivalent under D-2. (C) 2008 Elsevier B.V. All rights reserved.