$\beta$-ensembles and higher genera Catalan numbers

We propose formulas for the large $N$ expansion of the generating function of connected correlators of the $\beta$-deformed Gaussian and Wishart-Laguerre matrix models. We show that our proposal satisfies the known transformation properties under the exchange of $\beta$ with $1\mathrm{/}\beta$ and, using Virasoro constraints, we derive a recursion formula for the coefficients of the expansion. In the undeformed limit $\beta =1$, these coefficients are integers and they have the combinatorial interpretation of generalized Catalan numbers. For generic $\beta$, we define the higher genus Catalan polynomials $C_{g,\nu }(\beta )$ whose coefficients are integer numbers.