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

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