Interpretation of Array notation for Permutations EDIT: The big picture here is that I want to be able to reason about permutations in concrete terms. Most of the examples I see show permutations in either 2-row or cycle notation, but rarely refer back to the resulting configuration from applying them. In order to try to understand this better, I tried playing around with sequences of letters. It seemed natural to adapt the 2-row notation to refer instead to actual elements in a sequence, as in  $$\begin{pmatrix}\boldsymbol{A} & B & C\\\ \boldsymbol{2} & 1 & 3\end{pmatrix}$$. I realise this may have caused more problems than it solved.
It appears that a column in array notation for permutations expresses two different facts.
e.g.
$$\begin{pmatrix}\boldsymbol{A} & B & C\\\ \boldsymbol{2} & 1 & 3\end{pmatrix}$$

*

*The element which was in position 1 in the original goes to position 2 in the image


*The element in position 1 of the image is the element which came from position 2 in the original
I'm having trouble understanding this. Are the two facts somehow equivalent, like deriving facts about multiplication from facts about division? They seem somewhat unrelated to me, but that is maybe because I'm having trouble "getting my map oriented" when it comes to permutations.
Also, is there a term for the resulting configuration from applying a permutation? I.e. for the given example, the original sequence of $A, B, C$ would become $B, A, C$. How do we refer to $B, A, C$? Is it something like "the image of the original configuration under the permutation?"
Should this be described in terms of function composition? Maybe
$\left( \begin{array}{cc}
A & B & C \\
1 & 2 & 3
\end{array} \right)
%
\left( \begin{array}{cc}
A & B & C \\
2 & 1 & 3
\end{array} \right)
=
\left( \begin{array}{cc}
A & B & C \\
2 & 1 & 3
\end{array} \right)$
(I get that this is horribly wrong. I include it to illustrate the depth of my confusion. It doesn't help that I'm not sure if 2-row notation ever corresponds to a configuration rather than an action, and if so under what conditions.)
It seems the 2-row notation $$\begin{pmatrix}\boldsymbol{A} & B & C\\\ \boldsymbol{2} & 1 & 3\end{pmatrix}$$ actually contains the original configuration within the first row.
It's surprising to me how such an apparently simple notation can be so slippery. Can anyone please help me to clarify my understanding?
 A: Short answer that might help, offering a different notation.
I think I can see the two different ways you are thinking of a permutation: functions vs rearrangments. One way to think of permutations is a function, eg in the permutation below $\sigma(1) = 3, \sigma(2) = 4$, etc. In this example 1 changes to 3.
Another way to think about it is rearranging objects in front of you, eg if your favourite sports team where 1st last season, and are now third. In this example your favourite sports team doesn't change (it's still Mathletico Madrid, say), but it's position changes.
What I think you are trying to capture in your ABC123 notation is the second one, ie moving away from a function to a rearrangment. THere is something fixed changing position.

Here's a possible improvement on the ABC123 notation, try drawing coloured lines. Here you can see the red line (which is permenant), moving from 1st to 3rd.

You can also stack permutations vertically to get composition, and see where the red line ends up going under multiple permutations. Here's the function composed with it's inverse.

Hope this gives you some way of thinking about it that's clearer. (pictures taken from : https://groupsmadesimple.wordpress.com/2020/05/27/parity-of-permutations-by-pictures/ )
A: I'm not sure that this is what you look for, but you can label the elements of any finite set $A$ as $A=\{a_1,\dots,a_n\}$; then, any permutation $\sigma\in S_n$ induces a rearrangement of the elements of $A$ by means of the bijection on $A$ defined by $a_i\mapsto a_{\sigma(i)}$.
For example, call $a_1:=A, a_2:=B, a_3:=C$; then, the permutation $(123)$ (cycle notation) induces the rearrangement $(A,B,C)\mapsto (B,C,A)$.
