A permutation is a bijection from a set to itself:

Notation

Cauchy's two line notation maps each element of $S$ in the first row of a matrix to the element below it.

Example

Let $S=\{1,2,3,4,5,6\}$ and let $\sigma (1)=2,\sigma (2)=6,\sigma (3)=5,\sigma (4)=4,\sigma (5)=3,\sigma (6)=1$

We can write

If we view each mapping of $\sigma$ on a number line, with $V\to K$ when $K=\sigma (V)$:

\documentclass[tikz]{standalone}
\usetikzlibrary{arrows.meta}
%
\begin{document}
\begin{tikzpicture}
	\foreach \x in {1,2,3,4,5,6} {
	\node (\x) at (\x,0) {$\x$};}
	\begin{scope}[-Latex]
		\draw (1) edge[bend left] (2);
		\draw (2) edge[bend left] (6);
		\draw (6) edge[bend left] (1);
		\draw (3) edge[bend left] (5);
		\draw (5) edge[bend left] (3);
	\end{scope}
\end{tikzpicture}
\end{document}

we can reorder the list to convey disjoint cycles:

\documentclass[tikz]{standalone}
\usetikzlibrary{arrows.meta}
%
\begin{document}
\begin{tikzpicture}
	\foreach \x/\y in {1/1,2/2,3/4,4/6,5/5,6/3} {
	\node (\x) at (\y,0) {$\x$};}
	\begin{scope}[-Latex]
		\draw (1) edge[bend left] (2);
		\draw (2) edge[bend left] (6);
		\draw (6) edge[bend left] (1);
		\draw (3) edge[bend left] (5);
		\draw (5) edge[bend left] (3);
	\end{scope}
\end{tikzpicture}
\end{document}

which we call decomposition into disjoint cycles, and in cycle notation we would write

where any element $x\in S$ not appearing in any cycle, it is implicitly assumed that $\sigma (x)=x$.

Composition

Example

Let $$\sigma=\begin{pmatrix}
1 & 2 & 3 & 4 & 5 \
2 & 3 & 4 & 1 & 5
\end{pmatrix}
, \gamma=\begin{pmatrix}
1 & 2 & 3 & 4 & 5 \
3 & 1 & 5 & 4 & 2
\end{pmatrix}$$
Then $$\gamma\circ\sigma=\begin{pmatrix}
1 & 2 & 3 & 4 & 5 \
1 & 5 & 4 & 3 & 2
\end{pmatrix}$$
is the composition of $\gamma$ with $\sigma$, where the output of $\sigma$ is put into $\gamma$. (n.b. permutations are bijections, i.e. functions)