Crossings and nestings in four combinatorial families

The combinatorial families of matchings, set partitions, permutations and graphs can each be represented by a series of vertices along a horizontal line with arcs connecting them. Such a representation is referred to as an arc annotated sequence. A natural crossing and nesting structure arises in each of these representations, and remarkably enough, equidistribution between these two statistics has been shown for both matchings and partitions. To show this, tools such as RSK and several bijections are required. Furthermore, other useful bijections to lattice paths, and Ferrers diagrams give additional information, and aid the enumeration for each of the four families according to these two statistics.