Subfactorial

Subfactorial[n] gives the number of permutations of n objects that leave no object fixed (derangements).

Examples

Number of derangements:

Subfactorial[4]
(* 9 *)

Subfactorial sequence:

Table[Subfactorial[n], {n, 0, 6}]
(* {1, 0, 1, 2, 9, 44, 265} *)

Please visit the official Wolfram Language Reference for more details.