Pattern-avoiding permutations and Brownian excursion, part II: fixed points

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 fixed points of both 123- and 231-avoiding permutations. We find an exact description for a scaling limit of the empirical distribution of fixed points in terms of Brownian excursion. This builds on the connections between pattern-avoiding permutations and Brownian excursion developed in Hoffman et al. (Pattern-avoiding permutations and Brownian excursion, Part 1: Shapes and fluctuations. to appear Random Structures and Algorithms. arXiv:1406.5156 , 2016) and strengthens the recent results of Elizalde (Electron J Comb 18(2):17, 2011) and Miner and Pak (Adv Appl Math 55:86–130, 2014) on fixed points of pattern-avoiding permutations.