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 (2, ..., 2, 1) (2,..., 2, 2h + 1) π with π having length ℓ. This datum satisfies the Riemann-Hurwitz necessary condition for realizability if h — ℓ is odd and at least —1. For several small values of h and ℓ (namely, for h + ℓ ≤ 5) we explicitly compute the number v 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 v depends on arithmetic properties of the entries of π. In particular we find that in the only case where v is 0 the entries of π have a common divisor, in agreement with a conjecture of Edmonds-Kulkarny-Stong and a stronger one of Zieve.