理查森表与莫茨金路径
Richardson tableaux and Motzkin paths
- National Taiwan Normal University（台湾师范大学）
- National Pingtung University（屏东大学）
- National Cheng Kung University（国立成功大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文从莫茨金路径视角研究理查森表，引入独立于RS算法的形状算法，构造局部双射并重新证明q计数公式，还证明了郭关于特定理查森表生成函数的猜想。
AI中文摘要：
理查森表由卡普（Karp）和普雷库普（Precup）在研究作为理查森簇的施普林格纤维不可约分量时引入。郭（Guo）通过非交叉对合和罗宾逊-申斯特德（RS）算法，给出了理查森表与莫茨金路径之间的显式双射。本文从莫茨金路径的视角研究固定形状的理查森表，引入一种独立于RS算法的形状算法，可直接从莫茨金路径确定对应理查森表的形状。基于该算法，我们构造了一种局部双射，可追踪主统计量，并从组合角度重新证明卡普和普雷库普关于给定形状理查森表的q计数公式。我们还证明了郭关于具有指定奇数列数量的理查森表的次主生成函数的猜想。
英文摘要：
Richardson tableaux were introduced by Karp and Precup in their study of irreducible components of Springer fibers that are Richardson varieties. Guo gave an explicit bijection between Richardson tableaux and Motzkin paths through noncrossing involutions and the Robinson--Schensted (RS) algorithm. In this paper, we study Richardson tableaux of a fixed shape from the viewpoint of Motzkin paths. We introduce a shape algorithm, independent of the RS algorithm, that directly determines the shape of the corresponding Richardson tableau from a Motzkin path. Based on this algorithm, we construct a local bijection which keeps track of the major statistic and reproves combinatorially Karp and Precup's $q$-enumeration formula for Richardson tableaux of a given shape. We also prove in two ways a conjecture of Guo on the comajor generating function for Richardson tableaux with a prescribed number of odd columns.
