How to read permutation symbols like $(123)$? I'd be grateful for some help reading permutation symbols such as $(123)$. Does it mean, when applied to a target sequence such as $(x y z w)$, "replace the element in the first slot of the target with the element in the second slotof the target, the element in the second slot of the target with the element in the third slot of the target, and the element in the third slot of the target with the element in the first slot of the target," resulting in $(y z x w)$? If so, applied twice, $(123)$ would produce $(z x y w)$, which would mean that $(123).(123)=(321)$?
If I'm getting it right, then I should imagine little leftward arrows inside the permutation symbol, as in $(\leftarrow 1 \leftarrow 2 \leftarrow 3)$, with the first arrow implicitly wrapping around to the last position of the permutation symbol.
 A: The convention I'm familiar with is that $(123)$ means $1 \to 2 \to 3 \to 1$. Composition is just the usual composition of functions, and isn't described well by this example, so here's another example: $(12) \cdot (123) = (32)$. 
Applying permutations to sequences is tricky. Are you permuting the entries of the sequence, or the indices of the entries? One is a left action, and the other is a right action, of the symmetric group, and in the latter case there are inverses you need to add in to get a left action. 
A: Assuming $(1 2 3)$ refers to cycle notation then no, that's not the standard interpretation. It means "$1 to 2 to 3 to 1". Initially you shouldn't be thinking about the action of a permutation on some other set of symbols - think, more simply, about its action on the very symbols it comprises of (the numbers 1, 2, 3 in this case).
A more general permutation could look like $(1 4 5)(2 6)$ which means 1 goes to 4 who goes to 5 who goes back to 1, while 2 and 6 swap places and (implied) 3 stays put.
A: Generally $( 1 \ 2 \ 3)$ means, the elements $1 \to 2$ and $2 \to 3$ and $3 \to 1$. That is the element $1$ is sent to $2$ and $2$ is sent to $3$ and so on. Suppose you to operate $(1 \ 2  \ 3)$ with $(1 \ 3 \ 2)$ then we do it this way: since $1$ is sent to $2$ in $(1 \ 2 \ 3)$ we see where $2$ is sent in $(1 \ 3 \ 2)$. Now $2$ is sent to $1$ is in $(1 \ 3 \ 2)$ therefore $1$ is sent to $1$ in their product.
A: If $(123)$ is in "cycle notation", then this means that $1$ maps to $2$, $2$ maps to $3$, and $3$ (the last term in the cycle) maps to $1$. That is, those "little arrows" should point the other way. 
But with composition, you have to be careful! It depends on whether you are composing right-to-left (like functions), or left-to-right. For instance, if you write $(12)(13)$, then composing left-to-right (apply $(12)$ first, then $(13)$) you get $(123)$, but if you compose right-to-left, you get $(132)$. 
Which composition convention is being used depends on the author. But cycles are never, in my experience, read "right to left" themselves; that is, $(123)$ never represents $3\mapsto 2\mapsto 1\mapsto 3$. 
If $(123)$ is in "one line notation", then this would mean that the permutation is applied to a 3 element set, with $1$ mapping to $1$, $2$ mapping to $2$, and $3$ mapping to $3$ (i.e., $(abc)$ means $1\mapsto a$, $2\mapsto b$, $3\mapsto c$). However, one-line notation is not common. 
