AI 中文总结
该研究以避免特定vincular模式的排列为对象，借助莱默码将其转化为加权偏序集，证明其极大元由斐波那契数计数，且经典逆补映射在极大元间构成双射，揭示了偏序集结构的对称性。
AI 中文摘要
令$S_n(32\text{-}1)$和$S_n(3\text{-}21)$分别表示避免 vincular 模式$32\text{-}1$和$3\text{-}21$的n元排列集合。利用莱默码，我们将这些族实现为加权偏序集$L_n(32\text{-}1)$和$L_n(3\text{-}21)$，其中码的权重是其排列的逆序数。我们证明，每个偏序集的极大元$\text{Max}L_n(32\text{-}1)$和$\text{Max}L_n(3\text{-}21)$均由斐波那契数计数。我们证明，排列上的经典逆补映射限制在这两个极大元集合之间构成自然双射，揭示了它们底层偏序集结构间的深刻对称性。
英文摘要
Let $S_n(32\text{-}1)$ and $S_n(3\text{-}21)$ denote the sets of $n$-permutations avoiding the vincular patterns $32\text{-}1$ and $3\text{-}21$, respectively. Using Lehmer codes, we realize these families as weighted posets $L_n(32\text{-}1)$ and $L_n(3\text{-}21)$, where the weight of a code is the inversion number of its permutation. We show that the maximal elements of each of these posets, $\operatorname{Max} L_n(32\text{-}1)$ and $\operatorname{Max} L_n(3\text{-}21)$, are enumerated by the Fibonacci numbers. We demonstrate that the classical reverse-complement map on permutations restricts to a natural bijection between these two sets of maximal elements, revealing a deep symmetry between their underlying poset structures.
Comments23 pages, 5 figures
