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.
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, which requires certain estimates on characters of the symmetric group and uses combinatorics of Young tableaux. We conclude with conjectures and open problems.