Probability That A Random Subset of Lies On The Same Cycle
Select a random subset Q of containing m elements and a random permutation, and ask about the probability that all elements of Q lie on the same cycle. This is another average parameter. The function b(k) is equal to, because a cycle of length k contributes subsets of size m, where for k < m. This yields
Averaging out we obtain that the probability of the elements of Q being on the same cycle is
or
In particular, the probability that two elements p < q are on the same cycle is 1/2.
Read more about this topic: Random Permutation Statistics
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