Third order moments of complex Wigner matrices

Abstract: We compute the third order moments of a complex Wigner matrix. We provide a formula for the third order moments $\alpha_{m_1,m_2,m_3}$ in terms of quotient graphs $T_{m_1,m_2,m_3}^{\pi}$ 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.