One-line notation for permutations

Definition

Consider a finite set $S$ and an ordering of the elements of $S$, with the elements (in order), given as $a_1, a_2, \dots, a_n$. For a permutation $\sigma$ of $S$, the one-line notation for $\sigma$ is the string $\sigma(a_1) \ \sigma(a_2) \ \dots \ \sigma(a_n)$.

The one-line notation for a permutation is a compressed form for the two-line notation where the first line is omitted because it is implicitly understood.

For instance, for permutations on the set $\{ 1,2, \dots, n \}$ with the standard ordering, we can simply write the second line of the two-line notation where the first line is the standard ordering. For instance, consider a permutation $\sigma$ with two-line notation:

$\begin{pmatrix} 1 & 2 & 3 & 4 & 5 \\ 3 & 1 & 2 & 5 & 4 \\\end{pmatrix}$

Here, the first line is the standard ordering, and we can write the permutation using the one-line notation as:

$\begin{pmatrix}3 & 1 & 2 & 5 & 4 \\\end{pmatrix}$