  We introduce a notion of lexicographic shellability for pure, balanced
boolean cell complexes, modelled after the $CL$-shellability criterion of
Bj\"orner and Wachs for posets and its generalization by Kozlov called
$CC$-shellability. We give a lexicographic shelling for the quotient of the
order complex of a Boolean algebra of rank $2n$ by the action of the wreath
product $S_2\wr S_n$ of symmetric groups, and we provide a partitioning for the
quotient complex $\Delta (\Pi_n)/S_n $.
  Stanley asked for a description of the symmetric group representation
$\beta_S $ on the homology of the rank-selected partition lattice $\Pi_n^S $ in
[St2], and in particular he asked when the multiplicity $b_S(n)$ of the trivial
representation in $\beta_S$ is 0. One consequence of the partitioning for $\dps
$ is a (fairly complicated) combinatorial interpretation for $b_S(n) $; another
is a simple proof of Hanlon's result that $b_{1,..., i}(n)=0$. Using a result
of Garsia and Stanton, we deduce from our shelling for $\Delta (B_{2n})/S_2 \wr
S_n$ that the ring of invariants $k[x_1,..., x_{2n}]^{S_2\wr S_n}$ is
Cohen-Macaulay over any field $k$.
