Checking my work.
In either direction:
$(1243)[1] = 2$ and $(5)[2] = 2$, so far we have $(1, 2,\ldots$
$(1243)[2] = 4$ and $(5)[4] = 4$, so far we have $(1, 2, 4,\ldots$
$(1243)[4] = 3$ and $(5)[3] = 3$, so far we have $(1, 2, 4, 3, \ldots$
$(1243)[3] = 1$ and $(5)[1] = 1$, so we have $(1, 2, 4, 3).$
$(1243)[5] = 5$ and $(5)[5] = 5$, so we have $(5)$.
So, $(1243)(5) = (24135)$.
$(5)[1] = 1$ and $(1243)[1] = 2$, so far we have $(1, 2,\ldots$
$(5)[2] = 2$ and $(1243)[2] = 4$, so far we have $(1, 2, 4,\ldots$
$(5)[4] = 4$ and $(1243)[4] = 3$, so far we have $(1, 2, 4, 3,\ldots$
$(5)[3] = 3$ and $(1243)[3] = 1$, so we have $(1, 2, 4, 3)$.
$(5)[5] = 5$ and $(1243)[5] = 5$, so we have $(5)$.
So, $(1243)(5) = (24135)$.
Does that make sense?
edit:
I am trying to organize the topic of permutaions in my head. Here's what I wrote down(the question stems from it). Please, see what I need to correct:
Ways to represent permutations:
a) In ordered pairs form: $f = \{(x_1 y_1), (x_2 y_2), \ldots, (x_n y_n)\}$.
b) In $2 \times n $ matrix form: $f = \begin{pmatrix} x_1 & x_2 & \ldots & x_n \\ y_1 & y_2 & \ldots & y_n \end{pmatrix}$.
c) In cyclic form: $f = (x_1, f(x_1), f^2(x_1), \ldots, f^n(x_1)).$
d) In the form of values of the range of $f: f = (y_1, y_2, \ldots, y_n)$.
Example: $f = \underbrace{\{(12)(24)(31)(43)(55)\}}_{\text{(a)}} = \underbrace{\begin{pmatrix} 1 & 2 & 3 & 4 & 5 \\ 2 & 4 & 1 & 3 & 5 \end{pmatrix}}_{\text{(b)}} = \underbrace{(1243)(5)}_{\text{(c)}} = \underbrace{(24135)}_{\text{(d)}}$