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 $(2,\dots ,2)$, $(2h+1,1,2,\dots ,2)$, $\pi ={\left(d_i\right)}_{i=1}^{?}$, for several values of $g$ and $h$. We obtain explicit formulae of arithmetic nature in terms of the local degrees $d_i$. Our proofs employ a combinatorial method based on Grothendieck’s dessins d’enfant.