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:

$\left(\begin{array}{ccccc}1& 2& 3& 4& 5\\ 3& 1& 2& 5& 4\\ \end{array}\right)$

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

$\left(\begin{array}{ccccc}3& 1& 2& 5& 4\\ \end{array}\right)$