On enumeration of $b$-angulations of surfaces from an integrability perspective

In this paper, we study generating series enumerating polygonal angulations of closed oriented surfaces of fixed genus, focusing on $b$-angulations with $b=3$ or $b=2\nu$, $\nu \ge 2$. Based on Toda integrability, we establish new structural results in the cases $b=3$ and $b=4$. Furthermore, via the Hodge–GUE correspondence, we derive a fine structure in the $b=2\nu$ case, which implies a conjectural statement of Gharakhloo–Latimer.