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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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`.

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`.

Counting derangements uses subfactorials. For $ n $ items, the number of derangements is equal to $ !n $ (subfactorial of $ n $): $$ !n = n! \sum_{k=0}^n \frac {(-1)^k}{k!} $$

Derangements make exponential values. The more calculations there are, the more expensive are computer servers, so the large generations must be paid.

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*Derangements* on dCode.fr [online website], retrieved on 2023-02-08,

derangement,permutation,counting,subfactorial,fixed,point

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