Generalisations of the Harer–Zagier recursion for 1-point functions

Harer and Zagier proved a recursion to enumerate gluings of 2d-gon that result in an orientable genus g surface, their work on Euler characteristics moduli spaces curves. Analogous results have been discovered for other enumerative problems, so it is natural pose the following question: how large family problems which these so-called 1-point recursions exist? In this paper, we prove existence class Schur function expansions. particular, recover Harer–Zagier recursion, but our methodology also applies enumeration dessins d’enfant, Bousquet-Mélou–Schaeffer numbers, monotone Hurwitz more. On hand, there no governs single numbers. Our are effective sense one can explicitly compute particular instances recursions, provide several examples. We conclude paper with brief discussion conjecture relating theory topological recursion.