Reading cycle notation I need a little help understanding how we read cycle notation. I know it is a function so if we have something like
$\begin{pmatrix}1 & 2 & 3 & 4 & 5  \\
                2 & 5 & 4 & 3 & 1        
\end{pmatrix} = (1 \quad 2 \quad 5)(3 \quad 4)=(3 \quad 4)(1 \quad 2 \quad 5)=(3 \quad 4)(5\quad 1 \quad 2) $
how is it read? Thanks!

I think my original comment may have led to some confusion in regards to my intentions behind it--your function
\begin{pmatrix}1 & 2 & 3 & 4 & 5  \\\tag{1}
2 & 5 & 4 & 3 & 1        
\end{pmatrix}
may, perhaps, be more effectively communicated as
\begin{pmatrix}1 & 2 & 3 & 4 & 5  \\
\downarrow & \downarrow & \downarrow & \downarrow & \downarrow\\\tag{2}
2 & 5 & 4 & 3 & 1        
\end{pmatrix}
where each "$\downarrow$" in $(2)$ is really meant to be an upside down "$\mapsto$" (I'm not sure of the best way to rotate math symbols using MathJax). The point is that each number is being mapped to another. All of these mappings make up your function. Read aloud what happens in $(2)$:


*

*$1$ maps to $2$ which maps to $5$ which maps to $1$ which maps to $2$ which maps to $5$ ... 

*$3$ maps to $4$ which maps to $3$ which maps to $4$ which maps to $3$ ...


The point is that both of the cases above lead to noticeable cycles (hence the name). By understanding how the different mappings in $(2)$ work, we can now understand how the cycles were created, yielding
$$
\begin{pmatrix}1 & 2 & 3 & 4 & 5  \\
2 & 5 & 4 & 3 & 1        
\end{pmatrix}
=\begin{pmatrix} 1 & 2 & 5\end{pmatrix}\begin{pmatrix} 3 & 4\end{pmatrix}
$$
Does that help?
