Enumeration of unrooted hypermaps

In this paper we derive an enumeration formula for the number of hypermaps of given genus g and given number of darts n in terms of the numbers of rooted hypermaps of genus γ ⩽ g with m darts, where m | n . Explicit expressions for the number of rooted hypermaps of genus g with n darts were derived by Walsh (1975) for g = 0 , and by Arques (1987) for g = 1 . 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. The enumeration results can be expressed in terms of Fuchsian groups.