Combinatoric enumeration of two-dimensional proper arrays

An n×mn×mproper array   is a two-dimensional rectangular array composed of directed cubes that obey certain constraints. Because of these constraints, the n×mn×m proper arrays may be classified via a schema in which each n×mn×m proper array is associated with a particular n×1n×1 column. For a fixed n  , the goal is to enumerate, modulo symmetry, all possible edge configurations associated with n×mn×m proper arrays. By varying n  , one constructs four combinatoric sequences, each of which enumerates a particular class of edge configurations. Convolution arguments and resultant calculations associate these sequences with cubic equations. These cubic equations allow one to predict MnMn, the number of edge configurations, modulo symmetry, associated with n×mn×m proper arrays.