标准排列中的模式规避
Pattern avoidance in canon permutations
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中文总结 AI 辅助
本文研究任意k下标准排列的经典模式规避，证明规避指定单个模式的计数为k-卡特兰数，枚举相关组合类并建立与k元树的双射，推广相关定理，还利用k正则格词的规避问题枚举特定模式的标准排列并提出猜想。
中文摘要 AI 辅助
标准排列（canon permutation）是定义在集合[n]上的k正则词，其中对每个j，第j个字母的副本构成相同的排列σ。该概念由Elizalde引入，作为非嵌套多重排列的推广，当k=2时即为非嵌套多重排列。本文研究任意k下标准排列的经典模式规避问题，证明规避112、122、211或221中任意一个模式的计数为k-卡特兰数$\frac{1}{n}\binom{kn}{n-1}$；枚举了禁止上述模式之一与任意$\tau \notin \text{S}_3$组合得到的类，并给出其与k元树的双射，用于推广Gabriel、Peske、Pudwell和Tay的定理。随后证明规避在重标记下封闭的模式集合，其计数（除以因子n!后）可简化为k正则格词中的规避问题，利用该结果枚举了规避非嵌套与非交叉模式的自然推广、以及家族$\text{\textbraceleft}1^a21^b, 2^a12^b\text{\textbraceright}$的标准排列。最后提出若干猜想与问题。
英文摘要
A canon permutation is a $k$-regular word over $[n]$ in which, for each $j$, the $j$-th copies of the letters form the same permutation $σ$. These were introduced by Elizalde as a generalization of nonnesting multipermutations, which are the case $k = 2$. We study classical pattern avoidance in them for arbitrary $k$. We show that avoiding any one of $112$, $122$, $211$ or $221$ is counted by the $k$-Catalan numbers $\frac{1}{n}\binom{kn}{n-1}$. We enumerate the classes obtained by forbidding one of these together with any $τ\in \mathcal{S}_3$, and we give a bijection with $k$-ary trees that we use to generalize a theorem of Gabriel, Peske, Pudwell and Tay. We then show that avoiding a set of patterns closed under relabeling reduces, up to a factor of $n!$, to avoidance in $k$-regular lattice words. We use this to enumerate the canon permutations avoiding some natural generalizations of the nonnesting and noncrossing patterns, as well as the family $\{1^a21^b, 2^a12^b\}$. We close with several conjectures and questions.
