PATTERNS IN PERMUTATIONS AND INVOLUTIONS, A STRUCTURAL AND ENUMERATIVE APPROACH By CHEYNE HOMBERGER A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY

of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy PATTERNS IN PERMUTATIONS AND INVOLUTIONS, A STRUCTURAL AND ENUMERATIVE APPROACH By Cheyne Homberger May 2014 Chair: Miklós Bóna Major: Mathematics 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. This work is divided into five chapters. The first chapter introduces these techniques through working examples, both motivating the use of structural decomposition and showcasing the utility of their combination with analytic and probabilistic methods. The remaining chapters apply these concepts to separate aspects of permutation classes, deriving new enumerative, statistical, and structural results. These chapters are largely independent, but build from the same foundation to construct an overarching theme of structured disorder. The main results of this study are as follows. Chapter 2 investigates the average number of occurrences of patterns with permutation classes, and proves that the total number of 231-patterns is the same in the classes of 132and 123-avoiding permutations. Chapter 3 applies structural decomposition to enumerate pattern avoiding involutions. Chapter 4 uses the theory of grid classes to develop an algorithm to enumerate the so-called polynomial permutation classes, and applies this to the