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arXiv · 1405.2560

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.

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Jason P. Smith. 2015-07-30. Intervals of Permutations with a Fixed Number of Descents are Shellable. https://arxiv.org/abs/1405.2560

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