Realizations of certain odd-degree surface branch data

We consider surface branch data with base surface the sphere, odd degree d, three branching points, and partitions of d of the form with π having length l. This datum satisfies the Riemann-Hurwitz nec-essary condition for realizability if h-lis odd and at least 1. For several small values of h and l(namely, for h + l6 5) we explicitly compute the number π of realizations of the datum up to the equivalence relation given by the action of automorphisms (even unoriented ones) of both the base and the covering surface. The expression of π depends on arithmetic properties of the entries of π. In particular we find that in the only case where π is 0 the entries of π have a common divi-sor, in agreement with a conjecture of Edmonds-Kulkarny-Stong and a stronger one of Zieve.