Intervals of permutations with a fixed number of descents are shellable

Jason P. Smith

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Abstract

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.
Original languageEnglish
Pages (from-to)118–126
Number of pages9
JournalDiscrete Mathematics
Volume339
Issue number1
Early online date27 Aug 2015
DOIs
Publication statusPublished - 2015

Keywords

  • permutation Poset
  • shellability
  • möbius function

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