  Let (W,S) be a crystallographic Coxeter group (this includes all finite and
affine Weyl groups), and J a subset of S. Let $W^J$ denote the set of minimal
coset representatives modulo the parabolic subgroup $W_J$. For w in $W^J$, let
$f^{w,J}_{i}$ denote the number of elements of length i below w in Bruhat order
on $W^J$ (notation simplified to $f^{w}_{i}$ in the case when J=S). We show
that $f^{w,J}_{i}$ is less than or equal to $f^{w,J}_{j}$ when i < j and j is
less than or equal to the length of w minus i.
  Furthermore, we express when an initial and final interval of the f's is
symmetric around the middle in terms of Kazhdan-Lusztig polynomials.
  It is also shown that if W is finite then the sequence of f's cannot grow too
rapidly. Som result mirroring our first result are obtaind, again in the finite
case.
  The proofs rely for the most part on properties of the cohomology of
Kac-Moody Schubert varieties.
