  In this paper we show that counting Grothendieck's dessins d'enfants is
universal in the sense that some other enumerative problems are either special
cases or directly related to it. Such results provide concrete examples that
support a proposal made in the paper to study various dualities from the point
of view of group actions on the moduli space of theories. Connections to
differential equations of hypergeometric type can be made transparent from this
approach, suggesting a connection to mirror symmetry.
