Random monotone factorisations of the cycle

TL;DR

This paper explores monotone factorizations of cycles into transpositions, establishing a bijection with plane trees and examining their combinatorial and geometric properties.

Contribution

It introduces a novel bijection between monotone cycle factorizations and plane trees, linking combinatorics and geometry.

Findings

Bijection between monotone factorizations and plane trees

Connection to laminations as non-crossing line segments

Enhanced understanding of combinatorial properties of cycle factorizations

Abstract

In this article we study decreasing and increasing factorisations of the cycle, which are decompositions of the cycle $(1~2\dots n)$ into a product of $n-1$ transpositions satisfying monotonicity conditions. We explicit a bijection between such factorisations and plane trees with $n$ vertices. This will allow us to study some of their combinatorial properties, as well as a geometric representation in terms of laminations, which are non-crossing line segments in the unit disk.