Enumeration of unrooted hypermaps of a given genus

Abstract: In this paper we derive an enumeration formula for the number of hypermaps of a given genus and given number of darts in terms of the numbers of rooted hypermaps of genus with darts, where . Explicit expressions for the number of rooted hypermaps of genus with darts were derived by Walsh [T.R.S. Walsh, Hypermaps versus bipartite maps, J. Combin. Theory B 18 (2) (1975) 155–163] for , and by Arquès [D. Arquès, Hypercartes pointées sur le tore: Décompositions et dénombrements, J. Combin. Theory B 43 (1987) 275–286] for . We apply our general counting formula to derive explicit expressions for the number of unrooted spherical hypermaps and for the number of unrooted toroidal hypermaps with given number of darts. We note that in this paper isomorphism classes of hypermaps of genus are distinguished up to the action of orientation-preserving hypermap isomorphisms. The enumeration results can be expressed in terms of Fuchsian groups. [Copyright &y& Elsevier]