A good paper related to this question is P. Biane,  Representations of symmetric groups and free probability , Advances in Mathematics 138, 126-181 (1998); doi: [10.1006/aima.1998.1745](https://doi.org/10.1006/aima.1998.1745), item [4] at http://www.dma.ens.fr/~biane/publi.html ([Wayback Machine](https://web.archive.org/web/20080225042342/http://www.dma.ens.fr/~biane/publi.html)).