Enumeration of Set Partitions Refined by Crossing and Nesting Numbers

The standard representation of set partitions gives rise to two natural statistics: a crossing number and a nesting number. Chen, Deng, Du, Stanley, and Yan (2007) proved, via a non-trivial bijection involving sequences of Young tableaux that these statistics have a symmetric joint distribution. Recent results by Marberg (2013) has lead to algorithmic tools for the enumeration of set partitions with fixed crossing number and fixed nesting number. In this thesis we further consider set partitions refined by these two statistics. These sub-classes can be recognized by finite automata, and consequently have rational generating functions. Our main contribution is an investigation into the structure of the automata, the corresponding adjacency matrices, and the generating functions.