Triangulating the Permutahedron

For a finite irreducible real reflection group, $W$, we triangulate its permutahedron and use this to give an explicit homotopy equivalence between two known classifying spaces for the associated Artin group, $B(W)$. In the process, we characterise the Bruhat intervals from $w_1$ to $w_2$ in $W$, where $w_1^{-1}{w}_2$ is a Coxeter element.