Abstract
Branched covers of the complex projective line ramified over |$0,1,$| and |$\infty $| (Grothendieck's dessins d'enfant) of fixed genus and degree are effectively enumerated. More precisely, branched covers of a given ramification profile over |$\infty $| 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.
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