Peter Zograf has a new paper on the ArXiv entitled “Enumeration of Grothendieck’s Dessins and KP Hierarchy”.

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 overand 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 Dessinsof given genus and degree, thus providing a fast algorithm for computing these numbers.

You can download the paper here.