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Intervals of permutations with a fixed number of descents are shellable

Smith, Jason P. (2015) Intervals of permutations with a fixed number of descents are shellable. Discrete Mathematics, 339 (1). 118–126. ISSN 0012-365X

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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.