Abstract.Branched covers of the complex projective line ramified over
, and (Grothendieck’s Dessins d’Enfant) of fixed genus and degree are effectively enumerated. More precisely, branched covers of a given ramification profile over and given numbers of preimages of and are considered. The generating function for the numbers of such covers is shown to satisfy a PDE that determines it uniquely modulo a simple initial condition. Moreover, this generating function satisfies an infinite system of PDE’s called the KP (Kadomtsev-Petviashvili) hierarchy. A specification of this generating function for certain values of parameters generates the numbers of Dessins of given genus and degree, thus providing a fast algorithm for computing these numbers.
You can download the paper here.