Difference between revisions of "Derangement"
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==Notation== | ==Notation== | ||
| − | The number of derangements of an <math>n</math>-element set is called the <math>n</math>th derangement number or the ''subfactorial'' of <math>n</math> and is sometimes denoted <math>!n</math>. (Note that using this notation may require some care, as <math>a!b</math> can potentially mean both <math>(a!)b</math> and <math>a(!b)</math>.) This number is given by the formula | + | The number of derangements of an <math>n</math>-element set is called the <math>n</math>th derangement number or the ''subfactorial'' of <math>n</math> and is sometimes denoted <math>!n</math> or <math>D_n</math>. (Note that using this notation may require some care, as <math>a!b</math> can potentially mean both <math>(a!)b</math> and <math>a(!b)</math>.) This number satisfies the recurrences |
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| + | \[ | ||
| + | !n = n \cdot !(n - 1) + (-1)^n | ||
| + | \] | ||
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| + | and | ||
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| + | \[ | ||
| + | !n = (n - 1)(!(n - 1) + !(n - 2)) | ||
| + | \] | ||
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| + | and is given by the formula | ||
<cmath>!n = n! \sum_{k=0}^{n} \frac{(-1)^k}{k!}.</cmath> | <cmath>!n = n! \sum_{k=0}^{n} \frac{(-1)^k}{k!}.</cmath> | ||
Revision as of 22:43, 10 January 2008
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
The number of derangements of an 
-element set is called the th derangement 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
\[ !n = n \cdot !(n - 1) + (-1)^n \]
and
\[ !n = (n - 1)(!(n - 1) + !(n - 2)) \]
and is given by the formula
![\[!n = n! \sum_{k=0}^{n} \frac{(-1)^k}{k!}.\]](http://latex.artofproblemsolving.com/a/6/d/a6dc5b03b0f0635ae5384b8a78967ac7ae367890.png)
Thus, the number derangements of a 3-element set is 
, which we know to be correct.
Problems
Introductory
See also
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