Letter representations of m×n×p proper arrays

Let m 6= n. An m × n × p proper array is a three-dimensional rectangular array composed of directed cubes that obeys certain constraints. Because of these constraints, the m × n × p proper arrays may be classified via a schema in which each m×n×p proper array is associated with a particular m×n planar face. By representing each connected component present in the m × n planar face with a distinct letter, an m × n array of letters is formed. This m × n array of letters is the letter representation associated with the m × n × p proper array. The main result of this paper involves the enumeration of all m × n letter representations modulo symmetry, where the symmetry is derived from the group D2 = C2 × C2 acting on the set of letter representations. The 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 × n letter representations that are inequivalent under D2.