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

Anupam Chaudhuri, Norman Do

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Harer and Zagier proved a recursion to enumerate gluings of a 2d-gon that result in an orientable genus g surface, in their work on Euler characteristics of moduli spaces of curves. 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 this paper, 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 Bousquet-Mélou–Schaeffer numbers, to monotone Hurwitz numbers, and more. On the other hand, we prove that there is no 1-point recursion that governs single Hurwitz numbers. Our results are effective in the sense that one can explicitly compute particular instances of 1-point recursions, and we provide several examples. We conclude the paper with a brief discussion and a conjecture relating 1-point recursions to the theory of topological recursion.

Original languageEnglish
Pages (from-to)469-503
Number of pages35
JournalJournal of Algebraic Combinatorics
Issue number2
Publication statusPublished - Mar 2021


  • 1-point functions
  • Dessins d’enfant
  • Harer–Zagier formula
  • Holonomic functions
  • Hurwitz numbers
  • Ribbon graphs
  • Schur functions

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