The set of all permutations, ordered by pattern containment, forms a poset.
This paper presents the first explicit major results on the topology of
intervals in this poset. We show that almost all (open) intervals in this poset
have a disconnected subinterval and are thus not shellable. Nevertheless, there
seem to be large classes of intervals that are shellable and thus have the
homotopy type of a wedge of spheres. We prove this to be the case for all
intervals of layered permutations that have no disconnected subintervals of
rank 3 or more. We also characterize in a simple way those intervals of layered
permutations that are disconnected. These results carry over to the poset of
generalized subword order when the ordering on the underlying alphabet is a
rooted forest. We conjecture that the same applies to intervals of separable
permutations, that is, that such an interval is shellable if and only if it has
no disconnected subinterval of rank 3 or more. We also present a simplified
version of the recursive formula for the M\"obius function of decomposable
permutations given by Burstein et al.Comment: 33 pages, 4 figures. Incorporates changes suggested by the referees;
new open problems in Subsection 9.4. To appear in JCT(A