Patterns in Permutations and Involutions: A Structural and Enumerative Approach

TL;DR

This paper explores the structural and enumerative properties of permutation classes and involutions using geometric, analytic, and probabilistic methods to develop new enumeration techniques.

Contribution

It introduces a comprehensive framework combining geometric and probabilistic approaches for analyzing permutation classes and involutions.

Findings

New enumerative techniques for permutation classes

Structural decomposition methods for involutions

Integration of geometric and probabilistic analysis

Abstract

This dissertation presents a multifaceted look into the structural decomposition of permutation classes. The theory of permutation patterns is a rich and varied field, and is a prime example of how an accessible and intuitive definition leads to increasingly deep and significant line of research. The use of geometric structural reasoning, coupled with analytic and probabilistic techniques, provides a concrete framework from which to develop new enumerative techniques and forms the underlying foundation to this study.