The Wikipedia site on permutohedra displays a drawing of a 3-permutohedron in which vertices $\begin{pmatrix}2\\1\\3\end{pmatrix}$ and $\begin{pmatrix}3\\1\\2\end{pmatrix}$ are connected by an edge:
However, Günter Matthias Ziegler in "Lectures on Polytopes", ISBN 978-1-4613-8431-1, Example 0.10, p. 17, displays a drawing of a permutohedron $\Pi_2\subseteq\mathbb{R}^3$ with a labeling of the vertices in which the two aformentioned vertices are not connected by an edge:
Is Ziegler just wrong there or am I misinterpreting his drawing?
The two sources are using different notation.
Ziegler's vertex $(312)$ is a permutation that maps $x_3$ to $1$, $x_1$ to $2$, and $x_2$ to $3$, corresponding to the point $(2,3,1)$ in space. It is adjacent to $(132)$ and $(321)$ (Ziegler notation) which are points $(1,3,2)$ and $(3,2,1)$, all of which is consistent with the Wikipedia picture.