Combinatorial Structures and Generating Functions of Fishburn

This dissertation reflects the author's work on two problems involving combinatorial structures. The first section, which was also published in the Journal of Combinatorial Theory, Series A, discusses the author's work on several conjectures relating to the Fishburn numbers. The Fishburn numbers can be defined as the coefficients of the generating function begin{align*} 1+sum_{m=1}^{infty} prod_{i=1}^{m}(1-(1-t)^{i}). end{align*} Combinatorially, the Fishburn numbers enumerate certain supersets of sets enumerated by the Catalan numbers. We add to this work by giving an involution-based proof of the conjecture of Claesson and Linusson that the Fishburn numbers enumerate non-$2$-neighbor-nesting matchings. We begin by proving that a map originally defined by Claesson and Linusson gives a bijection between non-$2$-neighbor-nesting matchings and $textbf{(2-1)}$-avoiding inversion tables. We then define a set of diagrams, which we call Fishburn diagrams, that give a natural interpretation to the generating function of the Fishburn numbers. Using an involution on Fishburn diagrams, we then prove that the Fishburn numbers enumerate $textbf{(2-1)}$