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 finite critical values of which 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. The subject of the present paper is the structure of the algebras of transition types and approaches to efficient computation of simple real Hurwitz numbers.