# Intervals of permutations with a fixed number of descents are shellable

Jason P. Smith

Research output: Contribution to journalArticle

5 Citations (Scopus)

### 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 language English 118–126 9 Discrete Mathematics 339 1 27 Aug 2015 https://doi.org/10.1016/j.disc.2015.08.004 Published - 2015

Descent
Permutation
Interval
Poset
Subword
Bijection
Isomorphism
Alternatives

### Keywords

• permutation Poset
• shellability
• möbius function

### Cite this

Smith, Jason P. / Intervals of permutations with a fixed number of descents are shellable. In: Discrete Mathematics. 2015 ; Vol. 339, No. 1. pp. 118–126.
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In: Discrete Mathematics, Vol. 339, No. 1, 2015, p. 118–126.

Research output: Contribution to journalArticle

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KW - shellability

KW - möbius function

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