论文标题
shi tableaux和dyck路径中的图案
Patterns in Shi tableaux and Dyck paths
论文作者
论文摘要
Shi Tableaux是某些年轻图的特殊二元填充物，这些图是在与经典根系相关的SHI超平面布置研究中出现的。对于类型$ a $，shi tableaux的套装$ \数学t $自然与戴克路径相吻合，多年来已经引入和研究了各种图案的概念。在本文中，我们在$ \ Mathcal t $上定义了模式发生的概念，尽管它可以被视为戴克路径上的模式，但它是由tableaux的基本几何结构激发的。我们在这项工作中的主要目标是研究由图案含量定义的shi tableaux的poset。更确切地说，我们确定了\ Mathcal t $中每个$ T \的上下盖的明确公式，我们考虑最小的非平凡tableaux（尺寸2）的模式避免模式，并将这些结果推广到尺寸较大的某些平台。我们以空旷的问题和可能的未来方向结束。
Shi tableaux are special binary fillings of certain Young diagrams which arise in the study of Shi hyperplane arrangements related to classical root systems. For type $A$, the set $\mathcal T$ of Shi tableaux naturally coincides with the set of Dyck paths, for which various notions of patterns have been introduced and studied over the years. In this paper we define a notion of pattern occurrence on $\mathcal T$ which, although it can be regarded as a pattern on Dyck paths, it is motivated by the underlying geometric structure of the tableaux. Our main goal in this work is to study the poset of Shi tableaux defined by pattern-containment. More precisely, we determine explicit formulas for upper and lower covers for each $T\in\mathcal T$, we consider pattern avoidance for the smallest non-trivial tableaux (size 2) and generalize these results to certain tableau of larger size. We conclude with open problems and possible future directions.
