On partitions avoiding right crossings

TL;DR

This paper derives the generating function for set partitions avoiding right crossings by establishing a bijection with partial matchings avoiding 2-right crossings and right nestings, advancing combinatorial enumeration methods.

Contribution

It introduces a novel bijection between partial matchings avoiding 2-right crossings and right nestings and partitions avoiding right crossings, providing a new enumeration approach.

Findings

Derived the generating function for partitions avoiding right crossings.

Established a bijection linking partial matchings and partitions avoiding right crossings.

Connected avoidance of right crossings with other combinatorial structures.

Abstract

Recently, Chen et al. derived the generating function for partitions avoiding right nestings and posed the problem of finding the generating function for partitions avoiding right crossings. In this paper, we derive the generating function for partitions avoiding right crossings via an intermediate structure of partial matchings avoiding 2-right crossings and right nestings. We show that there is a bijection between partial matchings avoiding 2-right crossing and right nestings and partitions avoiding right crossings.