On Unfair Permutations

TL;DR

This paper investigates the properties of inverse unfair permutations, compares them with uniform permutations, and proves a central limit theorem for the number of inversions, introducing two generalizations of these permutations.

Contribution

It provides a detailed analysis of inverse unfair permutations, including their similarity to uniform permutations and the asymptotic distribution of the number of inversions.

Findings

Inverse unfair permutations behave similarly to uniform permutations for local dependencies.

The number of inversions in inverse unfair permutations satisfies a central limit theorem.

Two new generalizations of inverse unfair permutations are introduced.

Abstract

In this paper we study the inverse of so-called unfair permutations, and explore various properties of them. Our investigation begins with comparing this class of permutations with uniformly random permutations, and showing that they behave very much alike for locally dependent random variables. As an example of a globally dependent statistic we use the number of inversions, and show that this statistic satisfies a central limit theorem after proper centering and scaling. A secondary example of a globally dependent statistic to be studied will be the number of fixed points. Finally, we introduce two different generalizations of inverse-unfair permutations.