Mating of discrete trees and walks in the quarter-plane

TL;DR

This paper introduces a novel construction linking walks in the quarter plane to triangulations and maps with spanning trees, unifying various known bijections through a discrete mating of trees framework.

Contribution

It provides a general discrete mating of trees construction and a bijection between maps with spanning trees and quarter-plane walks, unifying multiple known bijections.

Findings

Established a new bijection between maps with spanning trees and quarter-plane walks.

Unified several existing bijections under a single framework.

Provided a discrete analog of the mating of trees concept.

Abstract

We give a general construction of triangulations starting from a walk in the quarter plane with small steps, which is a discrete version of the mating of trees. We use a special instance of this construction to give a bijection between maps equipped with a rooted spanning tree and walks in the quarter plane. We also show how the construction allows to recover several known bijections between such objects in a uniform way.