Derangement is a permutation without a fixed point.
Derangement - Wikipedia, the free encyclopedia
The probability that elements are deranged is given by . So the probability exhibits the following properties:
(1) When increases from an odd number to an even number (say, from 3 to 4), the probability increases. That is, when elements are shuffled, they get more likely to be deranged.
(2) When increases from an even number to an odd number (say, from 4 to 5), the probability decreases. That is, when elements are shuffled, they get more unlikely to be deranged.
(3) When is restricted to odd numbers, the probability is an increasing function of .
(4) When is restricted to even numbers, the probability is a decreasing function of .
I want, if any, "intuitive" explanations of these properties. Whether an explanation is "intuitive" is very subjective, so any ideas are welcome.
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