Derangement
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.
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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