# On pattern-avoiding Fishburn permutations

## TL;DR

This paper explores pattern avoidance in Fishburn permutations, providing enumerative results for various pattern classes, including those counted by Catalan and binomial transform numbers, and proposing conjectures for further classes.

## Contribution

It offers a complete enumeration of classical pattern avoidance in Fishburn permutations for size 3 and 4, introducing new enumerative results and conjectures for other pattern classes.

## Key findings

- Classical pattern avoidance in Fishburn permutations for size 3 fully characterized.
- Certain Fishburn permutation classes are enumerated by Catalan numbers.
- Additional classes are linked to binomial transforms of Catalan numbers.

## Abstract

The class of permutations that avoid the bivincular pattern (231, {1},{1}) is known to be enumerated by the Fishburn numbers. In this paper, we call them Fishburn permutations and study their pattern avoidance. For classical patterns of size 3, we give a complete enumerative picture for regular and indecomposable Fishburn permutations. For patterns of size 4, we focus on a Wilf equivalence class of Fishburn permutations that are enumerated by the Catalan numbers. In addition, we also discuss a class enumerated by the binomial transform of the Catalan numbers and give conjectures for other equivalence classes of pattern-avoiding Fishburn permutations.