Correspondence between the set of total orderings of a finite set $E$ and the symmetric group of $E$

Question

Given a finite set $E$, does there exist a canonical one-to-one correspondence between the set of total orderings of $E$ and $\mathfrak{S}_{E}$ (the symmetric group of $E$)?

Answer 1

Yes.
Let $<$ be a total order on $E$, eg if $E = \{x_1, \dots,x_n\}$, set $x_i<x_j :\Leftrightarrow i < j$.

A permutation $\sigma$ induces a total ordering $<_\sigma$ as follows :
$$ x <_\sigma y :\Longleftrightarrow \sigma^{-1}(x) < \sigma^{-1}(y)$$

And the correspondence $\varphi : \sigma \mapsto <_\sigma$ is a one to one correspondence between the permutations of $E$ and the total orderings on $E$ :

$\varphi$ is invertible:
Let $\prec$ be a total order on $E$. We build a permutation $\sigma =\varphi^{-1}(\prec)$ by induction :

• One sets $\sigma(1) := \min_\prec (E)$.
• Let $1<n<|E|$, if $\sigma(1),\dots,\sigma(n)$ were constructed, we set $\sigma(n+1) := \min_\prec E\setminus \{\sigma(1),\dots,\sigma(n)\}$