Tool for generating derangements. In mathematics, a derangement is a permutation of distinct objects without fixed point, ie that no object is in its original position.
Derangements - dCode
Tag(s) : Combinatorics
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Derangements
Derangements Generator
Counting Derangements
Answers to Questions (FAQ)
What is a derangement? (Definition)
The disturbances associated with a set of elements are a subset of its permutations. A fault is a permutation of elements without fixed points, that is to say without elements in the same position with the starting position of the whole.
Example: The set A,B,C has 2 faults C,A,B and B,C,A.
How to generate derangement?
The faster way to generate the list of derangements of a set is to list the permutations and remove those with fixed points (elements having an identical position in the permutation and in the starting position).
Example: The set A,B,C has 6 permutations: A,B,C B,A,C C,A,B A,C,B B,C,A C,B,A. Remove the one with fixed points, ie. the permutations with A in position 1, and/or those with B in position 2 and/or those with C in position 3.
The list of derangements are the 2 remaining permutations C,A,B and B,C,A.
How to count derangement?
Counting derangements uses subfactorials. For $ n $ items, the number of derangements is equal to $ !n $ (subfactorial of $ n $): $$ !n = n! \sum_{k=0}^n \frac {(-1)^k}{k!} $$
How to remove the limit when computing derangements?
Derangements make exponential values. The more calculations there are, the more expensive are computer servers, so the large generations must be paid.
Source code
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