A new family of bijections for planar maps

TL;DR

This paper introduces a novel bijective method for counting planar maps based on formulas from the KP hierarchy, providing the first bijective proof for quadratic map-counting formulas and expanding the combinatorial understanding of these structures.

Contribution

It presents a new bijection for planar maps that proves quadratic formulas from the KP hierarchy, extending beyond known linear cases.

Findings

First bijective proof for quadratic map-counting formulas from KP hierarchy

Introduction of a new 'cut-and-slide' bijection for planar maps

Bijection is not equivalent to existing ones, offering new combinatorial insights

Abstract

We present bijections for the planar cases of two counting formulas on maps that arise from the KP hierarchy (Goulden-Jackson and Carrell-Chapuy formulas), relying on a "cut-and-slide" operation. This is the first time a bijective proof is given for quadratic map-counting formulas derived from the KP hierarchy. Up to now, only the linear one-faced case was known (Harer-Zagier recurrence and Chapuy-F\'eray-Fusy bijection). As far as we know, this bijection is new and not equivalent to any of the well-known bijections between planar maps and tree-like objects.