> attempting to relay a rather simple concept

This notation is not for relaying the concept of permutations, for that (good) mathematics textbooks would simply write this explanation in plain English.

What the notation is good for is manipulating the notion of a permutation in a greater context, e.g. when you want to show that the functional composition of two permutations is itself a permutation, you can just replace this verbal description with the formal notation, shuffle around the symbols a bit, until you arrive at something that is obviously true.

Try doing that using the verbose English description.