Enumeration of Grothendieck's Dessins and KP Hierarchy

Branched covers of the complex projective line ramified over 0, 1, and infinity (Grothendieck's dessins d'enfant) of fixed genus and degree are effectively enumerated. More precisely, branched covers of a given ramification profile over infinity and given numbers of preimages of 0 and 1 are considered. The generating function for the numbers of such covers is shown to satisfy a partial differential equation (PDE) that determines it uniquely modulo a simple initial condition. Moreover, this generating function satisfies an infinite system of PDE's called the Kadomtsev-Petviashvili (KP) 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.