Bijections on two variations of noncrossing partitions

TL;DR

This paper establishes bijections among various combinatorial structures such as noncrossing partitions, Motzkin paths, and Schr{"o}der paths, revealing deep connections and providing direct mappings.

Contribution

It introduces new bijections between 2-distant noncrossing partitions and 12312-avoiding partitions, expanding the understanding of their combinatorial relationships.

Findings

Bijections between 2-distant noncrossing partitions and 12312-avoiding partitions

Bijections involving 3-Motzkin and Schr{"o}der paths

Direct correspondence between specific partition classes

Abstract

We find bijections on 2-distant noncrossing partitions, 12312-avoiding partitions, 3-Motzkin paths, UH-free Schr{\"o}der paths and Schr{\"o}der paths without peaks at even height. We also give a direct bijection between 2-distant noncrossing partitions and 12312-avoiding partitions.