Intervals of permutations with a fixed number of descents are shellable

The set of all permutations, ordered by pattern containment, is a poset. We present an order isomorphism from the poset of permutations with a fixed number of descents to a certain poset of words with subword order. We use this bijection to show that intervals of permutations with a fixed number of descents are shellable, and we present a formula for the Möbius function of these intervals. We present an alternative proof for a result on the Möbius function of intervals [1, π] such that π has exactly one descent. We prove that if π has exactly one descent and avoids 456123 and 356124, then the intervals [1, π] have no nontrivial disconnected subintervals; we conjecture that these intervals are shellable.