  We prove the conjecture of A. Postnikov that (A) the number of regions in the
inversion hyperplane arrangement associated with a permutation $w\in \Sn$ is at
most the number of elements below $w$ in the Bruhat order, and (B) that
equality holds if and only if $w$ avoids the patterns 4231, 35142, 42513 and
351624. Furthermore, assertion (A) is extended to all finite reflection groups.
  A byproduct of this result and its proof is a set of inequalities relating
Betti numbers of complexified inversion arrangements to Betti numbers of closed
Schubert cells. Another consequence is a simple combinatorial interpretation of
the chromatic polynomial of the inversion graph of a permutation which avoids
the above patterns.
