Intervals of permutations with a fixed number of descents are shellable

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• 作者：Jason P. Smith ^{jason.p.smith@strath.ac.uk" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Permutation poset ; Shellability ; Mö ; bius function
• 刊名：Discrete Mathematics
• 出版年：2016
• 出版时间：6 January 2016
• 年：2016
• 卷：339
• 期：1
• 页码：118-126
• 全文大小：404 K

文摘

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]$[1,\pi ]$ such that ac97cb7887e48198bc6a1732c5d" title="Click to view the MathML source">π$\pi$ has exactly one descent. We prove that if ac97cb7887e48198bc6a1732c5d" title="Click to view the MathML source">π$\pi$ has exactly one descent and avoids 456123 and 356124, then the intervals [1,π]$[1,\pi ]$ have no nontrivial disconnected subintervals; we conjecture that these intervals are shellable.