Combinatorics of Arc Diagrams, Ferrers Fillings, Young Tableaux and Lattice Paths

Several recent works have explored the deep structure between arc diagrams, their nestings and crossings, and several other combinatorial objects including permutations, graphs, lattice paths, and walks in the Cartesian plane. This thesis inspects a range of related combinatorial objects that can be represented by arc diagrams, relationships between them, and their connection to nestings and crossings. We prove a direct connection between nestings in involutions and the shape of Young tableaux, clarify Knuth transformations in terms of Young tableaux, present a local transformation on arc diagrams of involutions that we term involutive transformations, and describe variants to the well-known RSK correspondence.