  We introduce the Z-polynomial of a matroid, which we define in terms of the
Kazhdan-Lusztig polynomial. We then exploit a symmetry of the Z-polynomial to
derive a new recursion for Kazhdan-Lusztig coefficients. We solve this
recursion, obtaining a closed formula for Kazhdan-Lusztig coefficients as
alternating sums of multi-indexed Whitney numbers. For realizable matroids, we
give a cohomological interpretation of the Z-polynomial in which the symmetry
is a manifestation of Poincare duality.
