Enumeration of non-oriented maps via integrability

TL;DR

This paper explores how the BKP integrable structure can be used to derive explicit recurrence formulas for enumerating non-oriented maps, including triangulations and bipartite maps, with a focus on polynomial coefficient recurrences.

Contribution

It introduces methods to transform non-polynomial recurrence relations into polynomial ones for non-oriented map enumeration using BKP integrability techniques.

Findings

Derived recurrence formulas with polynomial coefficients for non-oriented maps.

Reproduced Ledoux's recurrence for non-oriented one-face maps.

Extended recurrence relations to bipartite maps.

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.