permutaatioksi

Permutaatioksi is the illative form of the Finnish noun permutaatiota about a permutation, used to describe something that is a permutation or transformed into a permutation. In mathematics, a permutation is a bijective map from a set to itself, which corresponds to a rearrangement of the elements of the set.

For a finite set with n elements, the total number of permutations is n!, and these permutations form the symmetric group S_n under composition. Permutations can be represented in several notations: one-line notation lists the images of the elements in order, cycle notation expresses disjoint cycles, and permutation matrices provide a matrix representation of the mapping. A permutation can be decomposed into disjoint cycles and can be expressed as a product of transpositions (two-element swaps), which shows how any permutation can be built from simple swaps.

Key properties include that permutations are structure-preserving bijections: they rearrange elements without duplicating or omitting any. They are invertible, and the inverse of a permutation is also a permutation. The concept of permutation is central to many areas of combinatorics, algebra, and computer science.

Applications range from sorting algorithms and data shuffling to cryptography, coding theory, and the study of symmetries in objects. In practice, a permutation describes how positions or elements are reordered, whether in arranging a list, shuffling a deck of cards, or modeling symmetry operations.

Example: On the set {1, 2, 3}, the permutation p defined by p(1) = 2, p(2) = 3, p(3) = 1 has one-line notation 2 3 1 and cycle notation (1 2 3). This illustrates how a permutation can be expressed in different, equivalent forms.