One-line notation for permutations

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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}