A recursion formula for mixed trace moments of isotropic Wishart matrices and the Gaussian unitary/orthogonal ensembles

Exact recursion formulas for mixed moments of four fundamental random matrix ensembles are derived. The reason such recursive formulas are possible is closely related to properties of polygon gluings studied by Harer and Zagier as well as Akhmedov and Shakirov. The proofs of the formulas are direct and written in such a way that they do not rely on understanding of polygon gluings.