Topological Recursion for Generalized $bc$-Motzkin Numbers

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.

arXiv comment46 pages, 12 figures, comments are welcome

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• Free energy topological expansion for the 2-matrix model
• Weil-Petersson volume of moduli spaces, Mirzakhani's recursion and matrix models
• Mirzakhani's recursion relations, Virasoro constraints and the KdV hierarchy
• The spectral curve of the Eynard-Orantin recursion via the Laplace transform
• Counting lattice points in compactified moduli spaces of curves