Convention on Cauchy's two line notation for permutations Let $n\in\mathbb{N}$. A permutation $\sigma\in S_n$ is denoted in Cauchy's two line notation as follow:
\begin{pmatrix}
1 & 2 & \cdots & n \\ \sigma(1) & \sigma(2) & \cdots & \sigma(n)
\end{pmatrix}
My question is how do we normally interpret $\sigma(i)$, for all $1\leq i\leq n$? I have found (at least) two different ways to interpret it:


*

*$\sigma(i)$ is the letter in the $i$th position (after the permutation is applied);

*$\sigma(i)$ is the position of the letter $i$ (after the permutation is applied).
In general these two interpretations give rise to different permutations. For example suppose $123$ is being rearranged into, says, $231$. Then interpretation 1 gives 
\begin{pmatrix}
1 & 2 & 3 \\ 2 & 3 & 1
\end{pmatrix}
but interpretation 2 gives 
\begin{pmatrix}
1 & 2 & 3 \\ 3 & 1 & 2
\end{pmatrix}
My question will be that which convention do mathematicians usually follow? 
Thanks in advance for clearing my doubt. 
 A: Neither of your interpretations are very clear, although interpretation 1 gives you the correct result if I understand you correctly.
A permutation acts on a set; thinking of the "position" of a letter after applying to permutation (to what?) does not make a lot of sense.  You seem to be thinking of having a string $123\ldots n$, where each letter has a distinct position, then you are applying the permutation to this string letter-by-letter to obtain a new string, where you can talk about the "positions".  This is a complicated viewpoint, and I don't know that it is especially useful.
Your first interpretation is closest to correct; you should modify it to say "$\sigma(i)$ is the letter obtained by applying the permutation to the letter $i$". Then if $123$ is arranged to $231$, you see that $\sigma:1 \to 2$, $\sigma:2 \to 3$, $\sigma:3 \to 1$. This gives you 
$$\begin{pmatrix} 1 & 2 & 3\\ \sigma(1) & \sigma(2) & \sigma(3)\end{pmatrix} = 
\begin{pmatrix} 1 & 2 & 3\\ 2 & 3 & 1\end{pmatrix}.$$
Notice the convenience of this notation; it is very easy to look up the image of any letter under your permutation.
A: Interpretation 1 is correct. If we are given a string ( not just a set ) we want to express permutations of the objects in some useful manner. That's what the Cauchy notation does.
" This is a complicated viewpoint, and I don't know that it is especially useful". As a physicist I assure you, it's the ONLY useful thing about permutation groups. I've never seen permutation groups in physics used any other way.
