Monotone leaky Hurwitz numbers

Develop monotone leaky Hurwitz numbers, including strict and weak monotone variants, and investigate whether they possess the structural properties known for monotone Hurwitz theories.

Background

Monotone and strictly monotone Hurwitz numbers have been shown in related work to exhibit structures such as chamber polynomiality, wall-crossing formulae, cut-and-join equations, topological recursion, integrability, and ELSV-type formulae. The paper has established several analogous properties for completed-cycle leaky Hurwitz numbers, but does not address monotonicity constraints.

The proposed problem is to construct and study monotone leaky Hurwitz numbers, determining whether the methods and structural results developed for ordinary and monotone Hurwitz theories extend to the leaky setting.