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:

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