Definition of Permutation in Mathematics 📂Lemmas

Definition of Permutation in Mathematics

Definition¹

A list that contains each element of a finite set $S$ exactly once is called a permutation of $S$. In particular, if the cardinality of $S$ is $|S| = n$ and the cardinality of a subset $T \subset S$ is $|T| = k$, then the number of possible permutations of $T$ is expressed as follows. $$ _{n} P _{k} = {\frac{ n! }{ (n - k) ! }} $$ Here, $n!$ is the $n$-factorial.

Easy Definition

The number of ways to choose $k$ out of $n$ distinct objects and arrange them in order is called a permutation and is denoted by $_{n}P_{k}$.

$$ {}_{n}P_{k} = n (n-1) \cdots (n-k+1) = \frac{ n! }{ (n-k)! } $$

Here, $n!$ is the factorial.

Explanation

A permutation is a concept that appears frequently throughout mathematics. Although the specific definitions may differ in details, it is basically viewed as a bijection that shuffles the array $[1, \cdots , n]$. When used in this way, it is often remarked that ‘only the order has been changed, so they are essentially the same’, and in such contexts it is easy to make the assumption during a proof that ‘without loss of generality, assume they are well sorted’.

Matrix Algebra

Definition of permutation matrix: A square matrix $P \in \mathbb{R}^{n \times n}$ in which each row has exactly one entry equal to $1$ and the rest equal to $0$ is called a permutation matrix.

Abstract Algebra

Definition of symmetric group: For a set $A$, a bijection $\phi : A \to A$ is called a permutation. $S_{A}$, the set of all permutations of $A$, forms a group $\left< S_{A} , \circ \right>$ under function composition $\circ$, and is called the symmetric group.