Combinatorial Patterns and Structures

这一建议的重点是在离散对象模式和结构的演变。通过一系列相互关联的问题，PI研究了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。通过代数组合学中的RSK算法的一种变体，匹配和集合分区与振荡和振荡表一一对应，振荡和振荡表是整数分区格的Hasse图中的某些随机游走。最近，在理解这类场景的统计数据方面取得了重大突破。在表示理论和随机矩阵理论等领域得到了许多意想不到的深刻联系。本文的研究主要集中在这些连接背后的组合性质，特别是在匹配和集合划分中的交叉和嵌套的性质，它们是排列中增减子序列的自然扩展。PI将从模式回避、Ferrer形状和生长图的填充、生成函数、渐近分布等方面研究问题。她还将理论扩展到一般图、和弦结构和某些多音的填充。在本研究中，一些算法在理解组合模式和结构的演化过程中起着重要作用。这种算法方法是该项目的特别重点。这项研究属于组合学的一般领域。组合学的目标之一是找到排列、枚举和操作离散对象集合的有效方法。针对这些目标提出的研究是正确的。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。建议的研究与数学的其他领域密切相关，包括分析、表示理论、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.