Enumeration of non-oriented maps via integrability

Abstract

In this note, we examine how the BKP structure of the generating series of several models of maps on non-oriented surfaces can be used to obtain explicit and/or efficient recurrence formulas for their enumeration according to the genus and size parameters. Using techniques already known in the orientable case (elimination of variables via Virasoro constraints or Tutte equations), we naturally obtain recurrence formulas with non-polynomial coefficients. This non-polynomiality reflects the presence of shifts of the charge parameter in the BKP equation. Nevertheless, we show that it is possible to obtain non-shifted versions, meaning pure ODEs for the associated generating functions, from which recurrence relations with polynomial coefficients can be extracted. We treat the cases of triangulations, general maps, and bipartite maps. These recurrences with polynomial coefficients are conceptually interesting but bigger to write than those with non-polynomial coefficients. However they are relatively nice-looking in the case of one-face maps. In particular we show that Ledoux's recurrence for non-oriented one-face maps can be recovered in this way, and we obtain the analogous statement for the (bivariate) bipartite case.

arXiv comment31 pages

References in corpus (18)

• Free energy topological expansion for the 2-matrix model
• D-particles, Matrix Integrals and KP hierachy
• Virasoro constraints and topological recursion for Grothendieck's dessin counting
• Counting colored planar maps: algebraicity results
• Local limits of uniform triangulations in high genus
• A short overview of the "Topological recursion"
• Weighted Hurwitz numbers and topological recursion
• Topological recursion for Kadomtsev-Petviashvili tau functions of hypergeometric type
• A bijection for rooted maps on general surfaces
• On irreducible maps and slices
• The geometry of spinors and the multicomponent BKP and DKP hierarchies
• $b$-monotone Hurwitz numbers: Virasoro constraints, BKP hierarchy, and $O(N)$-BGW integral
• The Non-Orientable Map Asymptotics Constant $p_g$
• Bijective enumeration of planar bipartite maps with three tight boundaries, or how to slice pairs of pants
• Topological recursion for generalised Kontsevich graphs and r-spin intersection numbers
• Simple formulas for constellations and bipartite maps with prescribed degrees
• Topological recursion for Orlov-Scherbin tau functions, and constellations with internal faces
• Blossoming bijection for bipartite pointed maps and parametric rationality of general maps of any surface

Cited by in corpus (2)

• Non-orientable branched coverings, $b$-Hurwitz numbers, and positivity for multiparametric Jack expansions
• Combinatorial proof for the rationality of the bivariate generating series of maps in positive genus