文摘
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 ac97cb7887e48198bc6a1732c5d" title="Click to view the MathML source">π has exactly one descent. We prove that if ac97cb7887e48198bc6a1732c5d" title="Click to view the MathML source">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.