Abstract
We establish a general bijective framework for encoding faces of some
classical hyperplane arrangements.
Precisely, we consider hyperplane arrangements in $\mathbb{R}^n$ whose
hyperplanes are all of the form $\{x_i-x_j=s\}$ for some $i,j\in[n]$ and $s\in
\mathbb{Z}$. Such an arrangement $A$ is \emph{strongly transitive} if it
satisfies the following condition: if $\{x_i-x_j=s\}\notin A$ and
$\{x_j-x_k=t\}\notin A$ for some $i,j,k\in [n]$ and $s,t\geq 0$, then
$\{x_i-x_k=s+t\}\notin A$.
For any strongly transitive arrangement $A$, we establish a bijection between
the faces of $A$ and some set of decorated plane trees.