Expected Number of Fixed Points in Random Permutation Raised To Some Power
Suppose you pick a random permutation and raise it to some power, with a positive integer and ask about the expected number of fixed points in the result. Denote this value by .
For every divisor of a cycle of length splits into fixed points when raised to the power Hence we need to mark these cycles with To illustrate this consider
We get
which is
Once more continuing as described in the introduction, we find
which is
The conclusion is that for and there are four fixed points on average.
The general procedure is
Once more continuing as before, we find
We have shown that the value of is equal to (the number of divisors of ) as soon as It starts out at for and increases by one every time hits a divisor of up to and including itself.
Read more about this topic: Random Permutation Statistics
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