Intuition behind multiplying (or composing) permutations. I'm trying to grasp the intuition for permutations and their multiplication. So far this has been my intuitive understanding: A permutation is merely a shuffling of the symbols. Take for example $\sigma , \pi \in S_4$ given by,
$ \sigma = \left(\begin{matrix}1 & 2 & 3 &4 \\ 3 & 2& 1& 4 \end{matrix}\right)$ and $ \pi= 
 \left(\begin{matrix}1 & 2 & 3 &4 \\ 2 & 4& 1& 3 \end{matrix}\right)$
Now from this answer, I could rewrite them as a $4-$tuple:
$\sigma = (3,2,1,4)$ and $\pi = (2,4,1,3)$ as permutations of $\{1,2,3,4\}$ and so $ \pi \circ \sigma = (2,4,1,3) \circ (3,2,1,4) = (1,4,2,3) \tag{#}$

I understand how to get the result. I know how to multiply (or compose) two permutations.

My Question: What happened in equation $\text{#}$ and what's going on intuitively? What shuffled around when composition happened? What does the result of product mean with respect to $\pi$ and $\sigma$?

 A: I'd say that it happens what the definition says it should happen: if you say that $f=(f_1,f_2,f_3,f_4)$ is the function such that $f(i)=f_i$, and if you say that $\pi\circ \sigma$ is the function such that $(\pi\circ \sigma)(i)=\pi(\sigma(i))$, then the representation of $\pi\circ\sigma$ will be given by $(\pi\circ\sigma)_i=\pi_{\sigma_i}$.
Of course, you should be careful whe you use this the one-line notation because you are using essentially the same notation as the very common cycle notation.
A: We can formulate the composition rule of permutations given in the form
\begin{align*}
\pi \circ \sigma = (2,4,1,3) \circ (3,2,1,4) = (1,4,2,3)
\end{align*}
as


*

*Select the items of the left-hand permutation $\pi$ in the order given by the items  of the right-hand permutation $\sigma$.


In the current example we have the order of the items of the right-hand permutation $\sigma$ as
\begin{align*}
\sigma=(3,2,1,4) \to \text{third, second, first and fourth item of }\pi
\end{align*}
resulting in
$$\color{blue}{\pi\circ\sigma=(1,4,2,3)}$$
