per·mu·ta·tion / ˌpərmyoŏˈtāshən/ • n. a way, esp. one of several possible variations, in which a set or number of things can be ordered or arranged: his thoughts raced ahead to fifty different permutations of what he must do. ∎  Math. the action of changing the arrangement, esp. the linear order, of a set of items.DERIVATIVES: per·mu·ta·tion·al / -ˈtāshənəl/ adj.

permutation of a set S. A bijection of S onto itself. When S is finite, a permutation can be portrayed as a rearrangement of the elements of S. The number of permutations of a set of n elements is n!

A permutation of the elements of {1,2,3} can be written 1 2 3 2 1 3

indicating that 1 is mapped into 2, 2 into 1, and 3 into 3. Alternatively the above can be written, using a cycle notation, as (1 2); this implies that the element 3 is unaltered but that 1 is mapped into 2 and 2 into 1.

For collections of elements in which repeated occurrences of items may exist, a permutation can be described as a rearrangement of elements in which each element appears with the same frequency as before.