In this dissertation, I show that a higher genus generalization of the bc-Motzkin numbers satisfies a topological recursion. These bc-Motzkin numbers are themselves a generalization of Catalan numbers, and, remarkably, the topological recursion satisfied by these generalized bc-Motzkin numbers is identical, up to a change of variable, to the topological recursion which had previously been proved for generalized Catalan numbers by Olivia Dumitrescu and Motohico Mulase. This topological recursion is an example of the Eynard-Orantin topological recursion, and it is one of multiple topological recursion formulas which have come from counting problems in combinatorics.
In the process of obtaining this topological recursion, I show that my higher genus generalization of bc-Motzkin numbers, which can be defined via an analogy with coloring graphs with v vertices on the genus g surface, satisfies a recursive formula, and, further, that the discrete Laplace transform of these numbers satisfies a differential recursion. It has been observed that the discrete Laplace transform of edge contraction operations in many graph counting problems corresponds to a topological recursion, and my example of the bc-Motzkin numbers is no exception.
Additionally, this definition of a higher genus generalization of bc-Motzkin numbers also leads to some identities and closed-form expressions for generalized Catalan numbers, in the case of small genus g and small number of vertices v. These results can be proved using the generating functions for these generalized bc-Motzkin numbers, and in this dissertation I show that these generating functions satisfy a recursive formula. Using the computer algebra system Mathematica to aid in computations, I obtain explicit generating functions for the generalized bc-Motzkin numbers for some cases of small (g,v). Then, based on these examples, I form a conjecture regarding the general form of the generating functions of these generalized bc-Motzkin numbers.