For example, what would be the quickest and easiest method to compute the composition $(1 3 2 4)\circ(1 4 2 3)$?
The method that I have been using is to convert it to two row notation, do the calculation, and then convert back to cyclic notation. I find this method cumbersome as it takes a large amount of space on my working out paper and it is rather long.
You can write the cycle decomposition in just one line. Rather than describe the algorithm, let me give an example which hopefully illustrates the idea.
Let us find the cycle decomposition for $(14)(345)(12345)$. For calculating this, here is how I would write it in one line, step by step.
(Note that I am using the convention that functions are written on the left.)
We start with the smallest number. Write: $(1$
Check $1 \mapsto 2$. Write: $(1 2$
Check $2 \mapsto 3 \mapsto 4 \mapsto 1$. Completed a cycle, write: $(12)$
Among the numbers that remain, smallest is $3$. Write: $(12)(3$
Next $3 \mapsto 4 \mapsto 5$. Write $(12)(35$
Then $5 \mapsto 1 \mapsto 4$. Write $(12)(354$
Then $4 \mapsto 5 \mapsto 3$, completed a cycle. Write $(12)(354)$
Conclude then that $(14)(345)(12345) = (12)(354)$.
To be clear, above by $2 \mapsto 3 \mapsto 4 \mapsto 1$ I mean the following: $2$ is mapped to $3$ by $(12345)$, then $3$ is mapped to $4$ by $(345)$, then $4$ is mapped to $1$ by $(14)$. Hence $2$ is mapped to $1$ by $(14)(345)(12345)$.