The classical Hurwitz enumeration problem has a presentation in terms of transitive factorizations in the symmetric group. This presentation suggests a generalization from type~$A$ to other finite reflection groups and, in particular, to type~$B$. We study this generalization both from a combinatorial and a geometric point of view, with the prospect of providing a means of understanding more of the structure of the moduli spaces of maps with an $\gS_2$-symmetry. The type~$A$ case has been well studied and connects Hurwitz numbers to the moduli space of curves. We conjecture an analogous setting for the type~$B$ case that is studied here.

MSC Classifications:

05A15, 14H10, 58D29 show english descriptions Exact enumeration problems, generating functions [See also 33Cxx, 33Dxx]
Families, moduli (algebraic)
Moduli problems for topological structures 05A15 - Exact enumeration problems, generating functions [See also 33Cxx, 33Dxx]
14H10 - Families, moduli (algebraic)
58D29 - Moduli problems for topological structures

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