Mixing and Covering in the Symmetric Groups

We discuss mixing and covering theorems in the symmetric groups. We present an example of a covering without mixing, and study the conjugacy class [2^n/2] of symmetric groupS_n, which demonstrates mixing without covering. We derive some new character identities from the computation of [2^n/2]², and also compute [2^n/2]³, filling a hole between theorems of J. L. Brenner (J. Austral. Math. Soc. Ser. A25,1978, 210-214) and Y. Dvir (in"Products of Conjugacy Classes in Groups," Lecture Notes in Math., Vol. 1112, Springer-Verlag, New York/Berlin, 1985). This computation also motivates a certain classification of 3-colored 3-regular graphs.