Symmetrically inequivalent partitions of a square array: part II.
Jocelyn Quaintance · Australas. J Comb. · 2009
An n × n × p proper array is a three-dimensional array composed of directed cubes that obeys certain constraints. Because of these constraints, the n× n× p proper arrays may be classified via a schema in which each n× n× p proper array is associated with a particular n× n planar face. By representing each connected component present in the n × n planar face with a distinct letter, and the position of each outward pointing connector by a circle, an n × n array of circled letters is formed. This n × n array of circled letters is the word representation associated with the n × n × p proper array. The main result of this paper involves the enumeration of all n× n word representations modulo symmetry, where the symmetry is derived from the group D4 = C4 × C2 acting on the set of word representations. This enumeration is achieved by forming a linear combination of six exponential generating functions, each of which is derived from a particular symmetry operation.