Word representations of m×n×pm×n×p proper arrays

Let m≠nm≠n. An m×n×pm×n×pproper array   is a three-dimensional array composed of directed cubes that obeys certain constraints. Due to these constraints, the m×n×pm×n×p proper arrays may be classified via a schema in which each m×n×pm×n×p proper array is associated with a particular m×nm×n planar face. By representing each connected component present in the m×nm×n planar face with a distinct letter, and the position of each outward pointing connector by a circle, an m×nm×n array of circled letters is formed. This m×nm×n array of circled letters is the word representation   associated with the m×n×pm×n×p proper array. The main result of this paper provides an upper bound for the number of all m×nm×n word representations modulo symmetry, where the symmetry is derived from the group D2=C2×C2D2=C2×C2 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 from a particular symmetry operation. This linear combination counts the number of partitions of the set of m×nm×n circled letter arrays that are inequivalent under D2D2.