Writing permutations as products of disjoint cycles (left to right)

Question

I've always done it from right to left but the book I'm currently using does it the other way around. Could someone please explain the process? Also I read here that answer for (1 2)(1 3) varies depending on the convention. I know there are many similar questions here but I can't say I understood any of the answers.

For example (1 2) (1 3) (1 4) and (4 5 2 1 3)(2 4 5 6)

Answer 1

Here is how it goes for the product of three transpositions:

• $1\mapsto 2$; the following transpositions leave $2$ fixed, so globally $\color{red}{1\mapsto 2}$.
• $2\mapsto 1$, then $1\mapsto 3$, which is fixed by the last transposition, so globally, $\color{red}{2\mapsto 3}$.
• $3\mapsto 3\mapsto 1\mapsto 4$, so $\color{red}{3\mapsto 4}$.
• $4\mapsto 4\mapsto 4\mapsto 1$, so $\color{red}{4\mapsto 1}$.

Therefore, we obtain the cycle $\enspace(1\:2\:3\:4)$.