The number of occurrences of patterns in a random tree or forest permutation

TL;DR

This paper investigates the distribution of pattern occurrences in random tree and forest permutations, demonstrating their asymptotic normality using U-statistics and limit theorems.

Contribution

It introduces a novel approach to analyze pattern counts in tree and forest permutations through U-statistics and asymptotic analysis.

Findings

Pattern occurrence counts are asymptotically normally distributed.

The method applies to permutations from classes defined by Acan and Hitczenko.

The approach leverages U-statistics and limit theorems for analysis.

Abstract

The classes of tree permutations and forest permutations were defined by Acan and Hitczenko (2016). We study random permutations of a given length from these classes, and in particular the number of occurrences of a fixed pattern in one of these random permutations. The main results show that the distributions of these numbers are asymptotically normal. The proof uses representations of random tree and forest permutations that enable us to express the number of occurrences of a pattern by a type of $U$-statistics; we then use general limit theorems for the latter.