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:

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