# The shuffle conjecture

## TL;DR

The paper discusses the shuffle conjecture, a significant combinatorial formula related to lattice walks and parking functions, highlighting its history, key developments, and algebraic proof by Carlsson and Mellit.

## Contribution

It provides an overview of the shuffle conjecture, detailing the combinatorial tools, historical context, and the algebraic solution, along with open problems.

## Key findings

- Algebraic proof by Carlsson and Mellit in 2018
- Historical development of the shuffle conjecture
- Remaining open problems in the field

## Abstract

Walks in the plane taking unit-length steps north and east from $(0,0)$ to $(n,n)$ never dropping below $y=x$ and parking cars subject to preferences are two intriguing ingredients in a formula conjectured in 2005, now famously known as the shuffle conjecture. Here we describe the combinatorial tools needed to state the conjecture. We also give key parts and people in its history, including its eventual algebraic solution by Carlsson and Mellit, which was published in the Journal of the American Mathematical Society in 2018. Finally, we conclude with some remaining open problems.