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Martin Sleziak
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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, item [4] at http://www.dma.ens.fr/~biane/publi.html (Wayback Machine).

Richard Stanley
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