Abstract :
A 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.
