Catalan polynomials for the Wishart-Laguerre $\beta$-ensemble

By means of $W$-representation, we investigate the Wishart-Laguerre $\beta$-ensemble and present a compact expression for Catalan polynomials. For a special Nekrasov partition function associated with the Wishart-Laguerre $\beta$-ensemble, we show that it can be expressed as a exponential form of the special Catalan polynomials. Furthermore, we make a rule to calculate the Catalan polynomials by means of Feynman diagrams and present another expression for Catalan polynomials. We also establish the connection between certain Catalan polynomials and knot theory.