We introduce two partially ordered sets, $P^A_n$ and $P^B_n$, of the same
cardinalities as the type-A and type-B noncrossing partition lattices.
The ground sets of $P^A_n$ and $P^B_n$ are subsets of the symmetric and the
hyperoctahedral groups, consisting of permutations which
avoid certain patterns. The order relation is given by
(strict) containment of the descent sets. In each case, by means of an
explicit order-preserving bijection, we show that the poset of restricted
permutations is an extension of the refinement order on noncrossing
partitions. Several structural properties of these permutation posets
follow, including self-duality and the strong Sperner property.
We also discuss posets $Q^A_n$ and $Q^B_n$ similarly associated with
noncrossing partitions.