| The secondary structure of RNA provides important information about the function of the RNA and its study is a fundamental problem in mathematical biology so there are a plethora of algorithms designed to predict a subset of possible RNA secondary structures. In this study, matchings are used to model RNA secondary structures, and describe the naturally defined families of matchings which arise from these algorithms. In addition, the study of patterns in matchings generalizes the study of permutation patterns. This dissertation generalizes an approach, known as the substitution decomposition, to the context of matchings and applies it to the study of RNA secondary structures. Chapter 1 introduces the biological context, outlines the substitution decomposition, and gives the enumeration of the H&S; family of matchings. Chapter 2 provides numerous examples of the enumeration of families of biological interest. Chapter 3 develops the general theory of the substitution to show that all hereditary families of matchings with only finitely many strongly indecomposables have algebraic generating functions and are well partially ordered. Chapter 4 gives the asymptotic enumerations of both weakly and strongly indecomposable matchings. |
