A Bijection for Crossings and Nestings

TL;DR

This paper introduces a direct bijection for a specific class of combinatorial objects that swaps maximum crossing and nesting numbers and all their refinements, highlighting limitations for coloured diagrams.

Contribution

It presents a novel bijection for certain matchings, set partitions, and permutations that interchanges crossing and nesting structures, extending understanding of their combinatorial properties.

Findings

Bijection interchanges maximum crossing and nesting numbers.

The bijection preserves all refinements of these numbers.

It cannot be applied to coloured arc annotated diagrams.

Abstract

For a subclass of matchings, set partitions, and permutations, we describe a direct bijection involving only arc annotated diagrams that not only interchanges maximum nesting and crossing numbers, but also all refinements of crossing and nesting numbers. Furthermore, we show that the bijection cannot be applied to a similar class of coloured arc annotated diagrams.