Abstract
Let
m≠n
. An
m×n×p
proper array is a three-dimensional 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 connencted 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 involves the enumeration 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 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
words representations that are inequivalent under
D
2
.