Permutations

#matematik-discmath

Definition: Permutation

A permutation of a non-empty finite set is a bijection .

We notice that by Theorem, it follows that the number of permutations of a set of size is .

Notation

For every we let

Theorem
  1. If , then .
  2. If , then .
  3. If there is an such that .

Proof:
Follows directly from the properties of bijections.

Definition: Cycle

A cycle is a permutation of given by where .

Notice that the starting number when denoting a cycle does not matter. Also note that every permutation can be written as the composition of cycles, such as