The Gini Index of an Integer Partition

TL;DR

This paper introduces a Gini index for integer partitions, linking it to symmetric polynomials and dominance order, and provides a generating function to analyze its properties and bounds within the lattice structure.

Contribution

It defines a novel Gini index for integer partitions, establishes its mathematical connections, and derives a generating function to study its combinatorial properties.

Findings

Gini index is related to the second elementary symmetric polynomial.

A generating function for the Gini index is derived.

The Gini index can be used to find lower bounds on the width of the dominance lattice.

Abstract

The Gini index is a number that attempts to measure how equitably a resource is distributed throughout a population, and is commonly used in economics as a measurement of inequality of wealth or income. The Gini index is often defined as the area between the Lorenz curve of a distribution and the line of equality, normalized to be between zero and one. In this fashion, we define a Gini index on the set of integer partitions and show that it is closely related to the second elementary symmetric polynomial, and the dominance order on partitions. We conclude with a generating function for the Gini index, and discuss how it can be used to find lower bounds on the width of the dominance lattice.