Word representations of proper arrays

Let m ≠ n . An m × n × p proper array is a three-dimensional array composed of directed cubes that obeys certain constraints. Due to 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, and the position of each outward pointing connector by a circle, an m × n array of circled letters is formed. This m × n array of circled letters is the word representation associated with the m × n × p proper array. The main result of this paper provides an upper bound for the number of all m × n word representations modulo symmetry, where the symmetry is derived from the group D 2 = C 2 × 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 from a particular symmetry operation. This linear combination counts the number of partitions of the set of m × n circled letter arrays that are inequivalent under D 2 .