Permutacija

Permutacija, or permutation, is an arrangement of objects in a specific order. In mathematics, a permutacija of a set is a bijection from the set to itself, equivalently a reordering of its elements. The number of permutations of n distinct elements is n!.

When elements are not all distinct, the number of distinct permutations changes. For a multiset with multiplicities n1, n2, ..., nk, the number of distinct permutations is n! divided by the product n1! n2! ... nk!. A k-permutacija from an n-element set is an ordered selection of k elements, with P(n, k) = n!/(n − k)! examples include the number of ways to choose and order k items from n.

Example: For the set {A, B, C}, there are 3! = 6 permutations: ABC, ACB, BAC, BCA, CAB, CBA. For the multiset {A, A, B}, there are 3! / 2! = 3 distinct permutations: AAB, ABA, BAA. A 2-permutacija from {A, B, C} yields P(3, 2) = 3 × 2 = 6.

In algebra, the set of all permutations of an n-element set forms the symmetric group S_n under composition. Every permutation can be decomposed into disjoint cycles, and this cycle decomposition is unique up to the order of the cycles. The identity permutation leaves every element fixed; its order is 1. A transposition swaps two elements, and any permutation can be expressed as a product of transpositions. The parity of a permutation is even or odd, depending on the number of transpositions in such a decomposition.