Increasing Forests and Quadrangulations via a Bijective Approach

TL;DR

This paper introduces four bijections that enable the transformation of forests and quadrangulations to alter their parameters, facilitating uniform sampling of these structures with different sizes.

Contribution

The work presents novel bijections that allow parameter adjustments in forests and quadrangulations, improving sampling methods for these combinatorial objects.

Findings

Enables sampling of quadrangulations with n+1 faces from those with n faces.

Allows generating forests with n+1 edges from those with n edges.

Provides a systematic method to increase or decrease parameters in these structures.

Abstract

In this work, we expose four bijections each allowing to increase (or decrease) one parameter in either uniform random forests with a fixed number of edges and trees, or quadrangulations with a boundary having a fixed number of faces and a fixed boundary length. In particular, this gives a way to sample a uniform quadrangulation with n + 1 faces from a uniform quadrangulation with n faces or a uniform forest with n+1 edges and p trees from a uniform forest with n edges and p trees.