Explicit computation of some families of Hurwitz numbers

We compute the number of (weak) equivalence classes of branched covers from a surface of genus g to the sphere, with 3 branching points, degree 2k, and local degrees over the branching points of the form (2,...,2), (2h+1,1,2,...,2), (d_1,...,d_m), for several values of g and h. We obtain explicit formulae of arithmetic nature in terms of the d_i's. Our proofs employ a combinatorial method based on Grothendieck's dessins d'enfant.
Comment: To appear in European Journal of Combinatorics (2018); 23 pages, 12 figures