Combinatorial Patterns and Structures

这个建议的重点是在离散对象的模式和结构的演变。通过一系列相互关联的问题，PI研究各种组合结构的性质和统计，包括排列，对合，匹配和集合划分。通过代数组合学中RSK算法的一个变体，匹配和集合划分与振荡和摇摆表一一对应，这些表是整数划分格的哈斯图中的某些随机游动。最近，在理解这类场景的统计数据方面有了重大突破。许多意想不到的和深刻的联系已经获得等领域的表示理论和随机矩阵理论。建议的研究集中在这些连接背后的组合属性，特别是，在匹配和集合分区，这是自然的扩展，增加和减少排列顺序的交叉和嵌套的属性。PI将从以下几个方面研究问题：模式避免，填充Ferrer形状和生长图，生成函数和渐近分布。她还将理论扩展到一般图形，弦配置和填充某些polyminos。在这项研究中，一些算法在理解组合模式和结构的演化方面起着重要的作用。这个演算法是这个计画的特别重点。这个研究是在组合学的一般领域。组合数学的目标之一是找到有效的方法来安排，枚举和操纵离散的对象集合。本研究正是针对这些目标而提出的。PI使用代数和算法相结合的方法来理解离散结构的形成和增长，这将有助于开发组合理论中的新方法和技术。所提出的研究与数学的其他领域密切相关，包括分析，表示论，随机矩阵理论和自由概率论。 该项目的成果将直接应用于计算机科学、电子工程、运筹学和统计学。

This proposal focuses on the evolution of patterns and structures in discrete objects. Through a series of interrelated problems, the PI investigates properties and statistics of various combinatorial structures, including permutations, involutions, matchings and set partitions. By a variant of the RSK algorithm from algebraic combinatorics, matchings and set partitions are in one-to-one correspondence with oscillating and vacillating tableaux, which are certain random walks in the Hasse diagram of the lattice of integer partitions. Recently there have been major breakthroughs in understanding the statistics of such tableaux. Many unexpected and deep connections have been obtained with such areas as representation theory and random matrix theory. The proposed research concentrates on the combinatorial properties behind these connections, in particular, on the properties of crossings and nestings in matchings and set partitions, which are natural extensions of increasing and decreasing subsequences in permutations. The PI will study the problems from the following aspects: pattern avoidance, filling of Ferrer shapes and growth diagram, generating functions, and asymptotic distribution.She will also extend the theory to general graphs, chord configurations, and filling of certain polyminos. In this research, some algorithm plays an important role in understanding the evolution of combinatorial patterns and structures. This algorithmic approach is a particular emphasis of the project.This research is in the general area of combinatorics. One of the goals of combinatorics is to find efficient methods for arranging, enumerating, and manipulating discrete collections of objects. The proposed research is right aimed at these goals. The PI uses a combined algebraic and algorithmic approach to understand the formation and growth of discrete structures, which will help develop new methods and techniques in combinatorial theory. The proposed research are closed connected to other areas of mathematics, including analysis, representation theory, random matrix theory, and free probability theory. Results from this project would have direct applications in computer sciences, electronic engineering, operation research, and statistics.