Difference between revisions of "Derangement"
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− | A '''derangement''' | + | A '''derangement''' is a [[permutation]] with no [[fixed point]]s. That is, a derangement of a [[set]] leaves no [[element]] in its original place. For example, the derangements of <math>\{1,2,3\}</math> are <math>\{2, 3, 1\}</math> and <math>\{3, 1, 2\}</math>, but <math>\{3,2, 1\}</math> is not a derangement of <math>\{1,2,3\}</math> because 2 is a fixed point. |
==Notation== | ==Notation== |
Revision as of 22:38, 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
[hide]Notation
The number of derangements of an -element set is called the th derangement number or the subfactorial of and is sometimes denoted . (Note that using this notation may require some care, as can potentially mean both and .) This number is given by the formula
Thus, the number derangements of a 3-element set is , which we know to be correct.
Problems
Introductory
See also
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