Subfactorial

The number of derangements of n objects: permutations that leave no element in its original position.

WikipediaDerangement↗Subfactorial↗MathWorldSubfactorial↗WikidataQ2361661↗OEISA000166↗Wolfram LanguageSubfactorial✓↗

Subfactorial(n)the number of derangements of n objects.

Domain: Combinatorics

Details

• Closed form , so is n! divided by e and rounded to the nearest integer.
• Recurrence: , with .
• Also satisfies the two-term recurrence .
• The classic hat-check problem: counts the ways n people can have their hats returned so nobody gets their own.
• compute-engine leaves negative arguments unevaluated rather than extending Subfactorial analytically.

Examples

See also: Factorial