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We investigate mixing of random walks on Sn and An generated by permutations of a given cycle structure. The approach follows methods developed by Diaconis; characters of the symmetric group and combinatorics of Young tableaux are used. We conclude with conjectures and open problems.
A hooklength formula for the number of rim hook tableaux is used to obtain an inequality relating the number of rim hook tableaux of a given shape to the number of standard Young tableaux of the same shape. This provides an upper bound for a certain family of characters of the symmetric group. The analogues for shifted shapes and rooted trees are also given.