Word Representations of m x n x p Proper Arrays

Let $m\neq n$. An $m\times n\times p$ proper array is a three-dimensional array composed of directed cubes that obeys certain constraints. Because of these constraints, the $m\times n\times p$ proper arrays may be classified via a schema in which each $m\times n\times p$ proper array is associated with a particular $m\times n$ planar face. By representing each connencted component present in the $m\times n$ planar face with a distinct letter, and the position of each outward pointing connector by a circle, an $m\times n$ array of circled letters is formed. This $m\times n$ array of circled letters is the word representation associated with the $m\times n\times p$ proper array. The main result of this paper involves the enumeration of all $m\times n$ word representations modulo symmetry, where the symmetry is derived from the group $D_2 = C_2\times C_2$ acting on the set of word representations. This enumeration 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 $m\times n$ words representations that are inequivalent under $D_2$.