具有经典和连续321避免的排列的最大Lehmer码
Maximal Lehmer Codes for Permutations with Classical and Consecutive 321-Avoidance
浏览论文内容
中文总结 AI 辅助
本文利用Lehmer码构造加权偏序集，研究避免经典和连续321模式的排列，确定了极大元素数量及最大权重元素的存在性，对应最大逆序数的排列。
中文摘要 AI 辅助
设${S}_n(321)$和${S}_n(\underline{321})$分别表示避免经典模式$321$和连续模式$\underline{321}$的$n$排列的集合。排列与Lehmer码（一种逆序列类型）之间存在双射。利用Lehmer码，我们构造了相应的加权偏序集${L}_n(321)$和${L}_n(\underline{321})$，其中码的权重是其排列的逆序数。我们证明了${L}_n(321)$有$2^{n-2}$个极大元素，而${L}_n(\underline{321})$的极大元素由Padovan数枚举。我们还证明了当$n$为偶数时，每个偏序集都有一个唯一的极大权重元素，当$n$为奇数时，存在两个极大权重元素。这些极大权重Lehmer码对应于具有最大逆序数的避免模式的排列。
英文摘要
Let ${S}_n(321)$ and ${S}_n(\underline{321})$ denote the sets of $n$-permutations avoiding the classical pattern $321$ and the consecutive pattern $\underline{321}$, respectively. Permutations are in bijection with Lehmer codes, a type of inversion sequence. Using Lehmer codes, we create the corresponding weighted posets ${L}_n(321)$ and ${L}_n(\underline{321})$, where the weight of a code is the inversion number of its permutation. We show that there are $2^{n-2}$ maximal elements of ${L}_n(321)$, while the maximal elements of ${L}_n(\underline{321})$ are enumerated by the Padovan numbers. We also show that when $n$ is even, each of these posets has a unique maximum weight element, and that when $n$ is odd, there are two maximum weight elements. These maximum weight Lehmer codes correspond to the pattern avoiding permutations with maximum inversion number.
发表机构
- Macalester College（麦克莱斯特学院）
机构由 AI 辅助整理，请以论文原文为准。
