Letter Representations of m x n x p Proper Arrays

Abstract

Let $m\neq n$. An $m\times n\times p$ {\it proper array} is a three-dimensional rectangular 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 connected component present in the $m\times n$ planar face with a distinct letter, an $m\times n$ array of letters is formed. This $m\times n$ array of letters is the {\it letter representation} associated with the $m\times n\times p$ proper array. The main result of this paper involves the enumeration of all $m\times n$ letter representations modulo symmetry, where the symmetry is derived from the group $D_2 = C_2\times C_2$ 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 $m\times n$ letter representations that are inequivalent under $D_2$. This is done by forming four generating functions, each of which derives from a particular symmetry operation.