  Let $W$ be a finite reflection group. For a given $w \in W$, the following
assertion may or may not be satisfied:
  (*) The principal Bruhat order ideal of $w$ contains as many elements as
there are regions in the inversion hyperplane arrangement of $w$.
  We present a type independent combinatorial criterion which characterises the
elements $w\in W$ that satisfy (*). A couple of immediate consequences are
derived:
  (1) The criterion only involves the order ideal of $w$ as an abstract poset.
In this sense, (*) is a poset-theoretic property.
  (2) For $W$ of type $A$, another characterisation of (*), in terms of pattern
avoidance, was previously given in collaboration with Linusson, Shareshian and
Sj\"ostrand. We obtain a short and simple proof of that result.
  (3) If $W$ is a Weyl group and the Schubert variety indexed by $w \in W$ is
rationally smooth, then $w$ satisfies (*).
