Topological Recursion for Generalized $bc$-Motzkin Numbers

TL;DR

This paper extends $bc$-Motzkin numbers to higher genus, deriving a recursive formula and demonstrating that their topological recursion aligns with the Eynard-Orantin framework previously applied to generalized Catalan numbers.

Contribution

It introduces a higher genus generalization of $bc$-Motzkin numbers and establishes their topological recursion as identical to that of generalized Catalan numbers.

Findings

Derived a recursive formula for higher genus $bc$-Motzkin numbers.

Showed the topological recursion matches the Eynard-Orantin recursion.

Connected $bc$-Motzkin numbers to established topological recursion frameworks.

Abstract

We present a higher genus generalization of $bc$-Motzkin numbers, which are themselves a generalization of Catalan numbers, and we derive a recursive formula which can be used to calculate them. Further, we show that this leads to a topological recursion which is identical to the topological recursion that had previously been proved by Dumitrescu and Mulase for generalized Catalan numbers, and which is an example of the Eynard-Orantin topological recursion.