Volumes of moduli spaces of directed ribbon graphs and Cut-and-Join operators

In this paper, we investigate the algebraic structure underlying the acyclic decomposition. This decomposition applies to directed metric ribbon graphs and enables the recursive computation of the volumes of their moduli spaces. Building on this, we define integral operators with these volumes and show that they satisfy a Cut-and-Join type equation. Furthermore, we demonstrate that a suitable specialization of these operators gives rise to a generating series for $\mathit{\text{dessins}}\mathit{\text{d’enfants}}$.