A reductive technique for enumerating non-isomorphic planar maps

AbstractA general method for the counting of unrooted planar maps is proposed. It reduces the problem to the counting of rooted maps of several classes of three kinds: planar, projective and ‘circular’. The latter are reduced further (for the set of all maps) to certain generalized rooted quadrangular dissections of the disc. Their counting in a ‘closed’ form remains so far an open problem.The method is based upon an exhaustive classification of the periodic homeomorphisms of the geometrical sphere, including orientation-reversing ones, into five types. A general formula of enumerating under orthogonal actions of a group is also derived