I will firstly introduce the concept of high order moments of a complex Wigner ensemble. The first order moments were studied by Wigner in the well-known Wigner’s law. The second order moments were studied by Male, Mingo, Péché and Speicher in their paper titled Joint Global Fluctuations of complex Wigner and deterministic Matrices, the main objective of this talk is to provide a formula for the third order moments. I will provide a formula for the third order moments $\alpha_{m_1,m_2,m_3}$ in terms of quotient graphs $T^{\pi}_{m_1,m_2,m_3}$ where $\pi$ is the Kreweras complement of a non-crossing pairing on the annulus. We prove that these graphs can be counted using the set of partitioned permutations, this permits us to write the third order moments in terms of the high order free cumulants which have a simple expression.

This is a joint work with James A. Mingo. This work was supported by a Discovery Grant from the Natural Sciences and
Engineering Research Council of Canada.