This article is the result of experiments performed using computer programs
written in the CAP language. We describe an algorithm which computes a set
of rational functions attached to a finite Coxeter group W. Conjecturally,
these rational functions should be polynomials, and in the case where W is
the Weyl group of a Chevalley group G defined over F-q, the values of our
polynomials at q should give the number of F-q-rational points of Lusztig's
special pieces in the unipotent variety of G. The algorithm even works for
complex reflection groups. We give a number of examples which show, in par
ticular, that our conjecture is true for all types except possibly B-n and
D-n.