Combinatorics of Bousquet-Mélou-Schaeffer numbers in the light of topological recursion

Abstract

In this paper we prove, in a purely combinatorial way, a structural quasi-polynomiality property for the Bousquet-Mélou-Schaeffer numbers. Conjecturally, this property should follow from the Chekhov-Eynard-Orantin topological recursion for these numbers (or, to be more precise, the Bouchard-Eynard version of the topological recursion for higher order critical points), which we derive in this paper from the recent result of Alexandrov-Chapuy-Eynard-Harnad. To this end, the missing ingredient is a generalization to the case of higher order critical points on the underlying spectral curve of the existing correspondence between the topological recursion and Givental's theory for cohomological field theories.

arXiv comment33 pages

References in corpus (20)

• Hurwitz numbers and intersections on moduli spaces of curves
• Remodeling the B-model
• Identification of the Givental formula with the spectral curve topological recursion procedure
• Virasoro constraints and topological recursion for Grothendieck's dessin counting
• Hypergeometric τ-functions, Hurwitz numbers and enumeration of paths
• Think globally, compute locally
• The spectral curve of the Eynard-Orantin recursion via the Laplace transform
• Reconstructing WKB from topological recursion
• Polynomiality of Hurwitz numbers, Bouchard-Mariño conjecture, and a new proof of the ELSV formula
• Multivariate hypergeometric functions as tau functions of Toda lattice and Kadomtsev-Petviashvili equation
• Dubrovin's superpotential as a global spectral curve
• Combinatorial solutions to integrable hierarchies
• Ramifications of Hurwitz theory, KP integrability and quantum curves
• Combinatorics of loop equations for branched covers of sphere
• Chiodo formulas for the r-th roots and topological recursion
• Combinatorial structure of colored HOMFLY-PT polynomials for torus knots
• Nonlinear PDEs for Fredholm determinants arising from string equations
• Polynomiality of orbifold Hurwitz numbers, spectral curve, and a new proof of the Johnson-Pandharipande-Tseng formula
• Towards an orbifold generalization of Zvonkine's $r$-ELSV formula
• Frobenius Manifolds near the Discriminant and Relations in the Tautological Ring

Cited by in corpus (8)

• Topological recursion for Kadomtsev-Petviashvili tau functions of hypergeometric type
• Explicit closed algebraic formulas for Orlov-Scherbin $n$-point functions
• Generalised ordinary vs fully simple duality for $n$-point functions and a proof of the Borot--Garcia-Failde conjecture
• Log topological recursion through the prism of $x-y$ swap
• Genus expansion of matrix models and $\hslash$ expansion of KP hierarchy
• Degenerate and irregular topological recursion
• The tensor Harish-Chandra-Itzykson-Zuber integral I: Weingarten calculus and a generalization of monotone Hurwitz numbers
• Taking limits in topological recursion