Letter Representations of m x n x p Proper Arrays

الأصل · EN

Let m≠ 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. This is done by forming four generating functions, each of which derives from a particular symmetry operation.