Random matrices and Hurwitz numbers

Many interesting counting problems are about factorizations in the symmetric group. A typical fundamental example is the Hurwitz numbers, originally counting covers of the complex projective line satisfying fixed ramification properties. Recently, new interesting variants of such Hurwitz problems have been introduced; in particular, monotone Hurwitz numbers and pruned Hurwitz numbers, both in single or double versions. Many of the properties of those numbers are not yet well understood and we want to explore those Hurwitz numbers and their interrelations from a tropical geometry and a free probability perspective.