Projects per year
Abstract
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 language | English |
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Pages (from-to) | 469-503 |
Number of pages | 35 |
Journal | Journal of Algebraic Combinatorics |
Volume | 53 |
Issue number | 2 |
DOIs | |
Publication status | Published - Mar 2021 |
Keywords
- 1-point functions
- Dessins d’enfant
- Harer–Zagier formula
- Holonomic functions
- Hurwitz numbers
- Ribbon graphs
- Schur functions
Projects
- 2 Finished
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Frobenius manifolds from a geometrical and categorical viewpoint
Norbury, P. T., Murfet, D. & Do, N.
2/03/18 → 31/12/20
Project: Research
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The geometry and combinatorics of moluli spaces
Australian Research Council (ARC)
30/06/13 → 30/08/18
Project: Research