# Catalan Intervals and Uniquely Sorted Permutations

## TL;DR

This paper explores the relationship between uniquely sorted permutations avoiding certain patterns and intervals in five posets on Dyck paths, using bijections, generating trees, and functions, and presents several conjectures.

## Contribution

It establishes equinumerosity between uniquely sorted pattern-avoiding permutations and intervals in five Dyck path posets, combining bijective and analytical methods.

## Key findings

- Uniquely sorted permutations avoiding patterns are equinumerous with poset intervals.
- Most proofs are bijective, some use generating trees and functions.
- Several conjectures are proposed for further research.

## Abstract

For each positive integer $k$, we consider five well-studied posets defined on the set of Dyck paths of semilength $k$. We prove that uniquely sorted permutations avoiding various patterns are equinumerous with intervals in these posets. While most of our proofs are bijective, some use generating trees and generating functions. We end with several conjectures.