Analysis of the Generalized Catalan Orbits

The Catalan numbers are well-known in the field of combinatorics. This sequence of integers counts a variety of combinatorial objects including binary trees and triangulations of regular polygons. Of the hundreds of realizations of the Catalan numbers, the majority can be extended and considered under the scope of the generalized Catalan numbers. The generalized Catalan numbers consider a second parameter in addition to the parameter of the regular Catalan numbers, consequently forming a grid of values instead of a sequence of integers. Although a formula has been developed to determine the generalized Catalan number when given values for the parameters, much is still unknown about the generalized Catalan numbers. This thesis will count the generalized Catalan orbits, defined to be the orbits of the combinatorial objects counted by the generalized Catalan numbers. Two realizations of the generalized Catalan numbers, p-ary trees and the dissection of regular polygons, will be used throughout the study of the generalized Catalan orbits. A variety of techniques will be used to create formulas which describe the number of generalized Catalan orbits in terms of the given parameters.