  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.
