# Lattice Paths and Pattern-Avoiding Uniquely Sorted Permutations

## TL;DR

This paper classifies and enumerates uniquely sorted permutations avoiding specific patterns by establishing bijections with lattice paths, thereby proving several conjectures in the field.

## Contribution

It introduces new enumeration results for pattern-avoiding uniquely sorted permutations and proves nine conjectures through bijective methods.

## Key findings

- Enumeration formulas for pattern-avoiding uniquely sorted permutations
- Bijections between permutations and lattice paths established
- Proof of nine conjectures by Defant

## Abstract

Defant, Engen, and Miller defined a permutation to be uniquely sorted if it has exactly one preimage under West's stack-sorting map. We enumerate classes of uniquely sorted permutations that avoid a pattern of length three and a pattern of length four by establishing bijections between these classes and various lattice paths. This allows us to prove nine conjectures of Defant.