Definition of Permutation in Mathematics
Definition 1
Explanation
A permutation is a concept that appears frequently throughout mathematics; while specific definitions may vary in detail, it is basically regarded as a bijection that shuffles the arrangement $[1, \cdots , n]$. In such usage it is often remarked that “only the order has been changed, fundamentally the same,” and in these contexts one commonly assumes in proofs the statement “without loss of generality, assume it is well-ordered.”
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 all other entries 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. $S_{A}$ is the set of all permutations of $A$, and under function composition $\circ$ it forms a group $\left< S_{A} , \circ \right>$, called the symmetric group.
Bóna, M. (2025). Introduction to enumerative and analytic combinatorics: p11. A list that contains each element of the 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 the 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 $n!$ denotes the $n$-factorial. ↩︎