Permutations and substitutions

A permutation of a set of $n$ distinct elements is a bijective function from the set to itself, meaning each element is mapped to exactly one other element, and every element is used exactly once. The number of permutations of $n$ objects is $n!$.

Important properties

• Permutations can be represented as rearrangements or as functions.

• The set of all permutations of $n$ elements forms the symmetric group $S_n$.

• Permutations can be written in two-line notation or cycle notation.

• Substitutions is another word for permutations, especially when thinking of them as functions.