On the structure of the algebra of transition types and the cut-and-join operator

Simple real Hurwitz numbers enumerate real meromorphic functions on real algebraic curves, all whose critical values are simple. M. Kazarian, S. Lando and S. Natanzon constructed algebras of transition types having these numbers as structure constants and deduced cut-and-join type equations for generating functions for them. In the present paper, we study the structure of the algebras of transition types and develop approaches to efficient computations of simple real Hurwitz numbers.