The Number of Ramified Coverings of the Sphere by the Torus and Surfaces of Higher Genera

I.P. Goulden^{1},D.M. Jackson^{1}, A. Vainshtein^{2}

{ipgoulden, dmjackso}@math.uwaterloo.ca

alek@cslx.haifa.ac.il

Annals of Combinatorics 4 (1) p.27-46 March, 2000

Abstract:

We obtain an explicit
expression for the number of ramified coverings of the sphere by the torus
with given ramification type for a small number of ramification points,
and conjecture this to be true for an arbitrary number of ramification
points. In addition, the conjecture is proved for simple coverings of the
sphere by the torus. We obtain corresponding expressions for surfaces of
higher genera for small number of ramification points, and conjecture the
general form for this number in terms of a symmetric polynomial that appears
to be new. The approach involves the analysis of the action of a transposition
to derive a system of linear partial differential equations that give the
generating series for the desired numbers.

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