Derangement (nonfiction)

All 24 permutations of 4 elements, with highlighting of the 9 derangements.

In combinatorial mathematics, a derangement is a permutation of the elements of a set, such that no element appears in its original position. In other words, derangement is a permutation that has no fixed points.

The number of derangements of a set of size n, usually written Dn, dn, or !n, is called the "derangement number" or "de Montmort number". (These numbers are generalized to rencontres numbers.) The subfactorial function (not to be confused with the factorial n!) maps n to !n.

No standard notation for subfactorials is agreed upon; n¡ is sometimes used instead of !n.

The problem of counting derangements was first considered by Pierre Raymond de Montmort in 1708; he solved it in 1713, as did Nicholas Bernoulli at about the same time.

Generalizations

Roblème des rencontres

The problème des rencontres asks how many permutations of a size-n set have exactly k fixed points.

Anagrams

Another generalization is the following problem:

How many anagrams with no fixed letters of a given word are there?

Fiction cross-reference

• Crimes against mathematical constants
• Gnomon algorithm
• Gnomon Chronicles
• Mathematician
• Mathematics

Nonfiction cross-reference

• Mathematics (nonfiction)
• Nicholas Bernoulli (nonfiction)
• Pierre Raymond de Montmort (nonfiction)
• Rencontres numbers (nonfiction) - a triangular array of integers that enumerate permutations of the set { 1, ..., n } with specified numbers of fixed points: in other words, partial derangements.

External links:

• Derangement @ Wikipedia