Definition:Derangement

Definition
A derangement is a permutation $f: S \to S$ from a set $S$ to itself where $\map f s \ne s$ for any $s \in S$.

If $S$ is finite, the number of derangements is denoted by $D_n$ where $n = \card S$ (the cardinality of $S$.)

Also see

 * Recurrence Relation for Number of Derangements on Finite Set
 * Closed Form for Number of Derangements on Finite Set
 * Definition:Subfactorial


 * Hat-Check Problem