Each conjugacy class of S_n is determined by the cycle structure of its elements. A quick way to find a random cycle structure with probability proportional to the size of the corresponding class is as follows. If it is only the cycle structure which you need this is faster than finding a random element of S_n.

   Set m := n and clist := []
   While m > 0 do
      {Choose  k  (uniformly) at random from [1..m]
      Add k to the clist and set m  := m-k}
   return clist

   The resulting clist should be read as a list of cycle lengths.
[ See Dixon and Mortimer, "Permutation Groups", Springer 1996, Exercise 3.6.6.]

      - John Dixon
Don King wrote:

I`d like to sample some elements in a symmetric group of order n based on the ratio of conjugacy classes.

For instance, if a symmetric group just has five conjugacy classes (this is just for illustration, not an actual symmetric group),

Class 1: 10 memeber
Class2: 20 members
Class3: 30 memebers
Class4: 20 members
Class5: 10 memebers

is it possible to pick random samples like 1,2,3,2,1 elements from each class based on the ratio of each conjugacy classes ?.

Is there any way to pick random samples from each conjugacy class ?.

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