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...

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Bibliographic Details
Published in:arXiv.org 2015-07
Main Author: Smith, Jason P
Format: Article
Language:English
Subjects:
Containment
Descent
Intervals
Isomorphism
Permutations
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Summary: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 M\"obius function of these intervals. We present an alternative proof for a result on the M\"obius function of intervals \([1,\pi]\) such that \(\pi\) has exactly one descent. We prove that if \(\pi\) has exactly one descent and avoids 456123 and 356124, then the intervals \([1,\pi]\) have no nontrivial disconnected subintervals; we conjecture that these intervals are shellable.
ISSN:2331-8422