Derangement
From AoPSWiki
A derangement is a permutation with no fixed points. That is, a derangement of a set leaves no element in its original place. For example, the derangements of 
are 
and 
, but 
is not a derangement of because 2 is a fixed point.
Contents |
Notation and formula
The number of derangements of an 
-element set is called the th derangement number or rencontres number, or the subfactorial of and is sometimes denoted 
or 
. (Note that using this notation may require some care, as 
can potentially mean both 
and 
.) This number satisfies the recurrences
and
and is given by the formula
For example, the number derangements of a 3-element set is 
, which we know to be correct.
The first few derangements, starting from 
, are 
Limit as n approaches ∞
From the recurrence 
, we can find that 
Also, 
Problems
Introductory
See also
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