On Inversion Graphs of Permutations

TL;DR

This paper investigates the structure of inversion graphs derived from permutations, connecting combinatorics, geometry, and poset theory to uncover their complex properties.

Contribution

It provides a comprehensive analysis of inversion graphs of permutations, integrating multiple perspectives to deepen understanding of their combinatorial structure.

Findings

Characterization of inversion graphs in various contexts

Connections between inversion graphs and geometric/poset structures

Insights into the combinatorial properties of permutation inversion graphs

Abstract

In this dissertation, we explore the structure of inversion graphs of permutations--a class of graphs that naturally arises by representing each permutation as a graph, where vertices correspond to entries and edges encode inversions. Advantageously, this class also arises from several other definitions that relate these graphs to geometry and partially ordered sets. By leveraging these different perspectives, we are equipped to gain many insights into their combinatorial intricacies.