  We introduce a new statistic on the hyperoctahedral groups (Coxeter groups of
type B), and give a conjectural formula for its signed distributions over
arbitrary descent classes. The statistic is analogous to the classical Coxeter
length function, and features a parity condition. For descent classes which are
singletons the conjectured formula gives the Poincar\'e polynomials of the
varieties of symmetric matrices of fixed rank.
  For several descent classes we prove the conjectural formula. For this we
construct suitable "supporting sets" for the relevant generating functions. We
prove cancellations on the complements of these supporting sets using suitably
defined sign reversing involutions.
