Pattern-avoiding permutations and Brownian excursion Part I: Shapes and fluctuations

TL;DR

This paper investigates the scaling limits of pattern-avoiding permutations of length 3 and their relation to Brownian excursions, providing new insights into their shape fluctuations and strengthening recent theoretical results.

Contribution

It establishes a connection between pattern-avoiding permutations and Brownian excursions, enhancing understanding of their asymptotic behavior and fluctuations.

Findings

Permutations avoiding length-3 patterns converge to Brownian excursion shapes.

The study clarifies the fluctuations and limit shapes of these permutations.

Strengthens previous results by Madras, Pehlivan, Miner, and Pak.

Abstract

Permutations that avoid given patterns are among the most classical objects in combinatorics and have strong connections to many fields of mathematics, computer science and biology. In this paper we study the scaling limits of a random permutation avoiding a pattern of length 3 and their relations to Brownian excursion. Exploring this connection to Brownian excursion allows us to strengthen the recent results of Madras and Pehlivan, and Miner and Pak as well as to understand many of the interesting phenomena that had previously gone unexplained.