Crossings and nestings in set partitions of classical types

TL;DR

This paper explores bijections in set partitions of classical types, focusing on crossings and nestings, and extends known results from type A to types B, C, and D, with applications to triangulations and Dyck paths.

Contribution

It generalizes bijections between crossings and nestings from type A to types B, C, and D, and addresses a conjecture related to type B triangulations and Dyck paths.

Findings

Bijections that interchange crossings and nestings for types B and C

A bijection between non-crossing and non-nesting partitions for type D

Resolution of a conjecture on type B triangulations and symmetric Dyck paths

Abstract

In this article, we investigate bijections on various classes of set partitions of classical types that preserve openers and closers. On the one hand we present bijections that interchange crossings and nestings. For types B and C, they generalize a construction by Kasraoui and Zeng for type A, whereas for type D, we were only able to construct a bijection between non-crossing and non-nesting set partitions. On the other hand we generalize a bijection to type B and C that interchanges the cardinality of the maximal crossing with the cardinality of the maximal nesting, as given by Chen, Deng, Du, Stanley and Yan for type A. Using a variant of this bijection, we also settle a conjecture by Soll and Welker concerning generalized type B triangulations and symmetric fans of Dyck paths.