Enumeration and Structure in Families of Partitions, Compositions, and Combinations

分区、组合和组合族中的枚举和结构

• 批准号: 0300034
• 负责人: Carla Savage
• 金额: $ 18.33万
• 依托单位: North Carolina State University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2003
• 资助国家: 美国
• 起止时间: 2003-07-15 至 2007-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0300034&HistoricalAwards=false
• 关键词: Enumeration; Structure; Families; Partitions; Compositions

Abstract for award DMS-0300034 of SavageThe proposed research is an investigation of fundamental questionsinvolving the structure of combinatorial families and relationshipsbetween families with intrinsically different characterizations.The first part concerns partitions and compositions constrained by linear inequalities. Recent research has shown this framework to provide a common setting for many partition identities. It has produced surprising connectionswith families defined by rank conditions, by forbidden parts, and by difference conditions. The PI seeks to identify the families that can be characterized inthis way, the partition statistics that can be captured, and the new insight that might be gained. The second part of the research studies identitiesof the Rogers-Ramanujan type. There has been a growing recognition of the importance of these identities in statistical physics andLie algebra and, as a result, an explosion of research uncovering new identities of the Rogers-Ramanujan type. Nevertheless, these identities are still not well understood combinatorially. The PI investigates new tools to analyze the combinatorial aspects. The third part of the project focuses on structure in partially ordered sets, specifically, symmetric chaindecompositions. This extends recent work of the PI and colleagues that used symmetric chain decompositions to solve an open geometric question about the existence of symmetric Venn diagrams. It pursues a new approach to the outstanding open question of the existence of symmetric chain decompositionin certain important posets.Combinatorics is the mathematics used to investigate, analyze, and manipulatestructured data sets: the pages of the world-wide web, the nucleotides forming DNA, the customers in a telephone network, or the configuration ofsubatomic particles in the nucleus of atoms. Combinatorics underlies criticalcomputer algorithms for retrieving information, for designing communication networks, for encrypting transactions, and for sequencing DNA. It is of economic and strategic importance to have a scientific workforce with expertise in this critical area, which has yet to enter the traditional public school curriculum. The investigator of this project is committed to the training and involvement of students in all aspects of the research. It is the nature of the work that the compelling open questions attract students at both the undegraduate and graduate level, many of whomhave made substantial contributions in previous projects with this P.I.The results of this project will be useful to other areas of mathematics, such as ordered sets and representation theory of Lie groups, to the study of the statistical behavior of bosons and fermions in lasers and superconductors, and to the visualization of data in statistics.

Savage DMS-0300034 奖摘要所提出的研究是对涉及组合族结构以及具有本质上不同特征的族之间关系的基本问题的调查。第一部分涉及受线性不等式约束的划分和组合。 最近的研究表明，该框架为许多分区身份提供了通用设置。它与由等级条件、禁忌部分和差异条件所定义的家庭建立了令人惊讶的联系。 PI 旨在确定可以通过这种方式表征的系列、可以捕获的分区统计数据以及可能获得的新见解。 研究的第二部分研究罗杰斯-拉马努金类型的身份。人们越来越认识到这些恒等式在统计物理学和李代数中的重要性，因此，大量研究发现了罗杰斯-拉马努金类型的新恒等式。 然而，这些恒等式的组合仍然没有得到很好的理解。 PI 研究新工具来分析组合方面。 该项目的第三部分重点关注部分有序集合中的结构，特别是对称链分解。 这扩展了 PI 及其同事最近的工作，他们使用对称链分解来解决有关对称维恩图存在的开放几何问题。 它寻求一种新的方法来解决某些重要偏序集中对称链分解的存在这一悬而未决的问题。组合学是用于研究、分析和操作结构化数据集的数学：万维网的页面、形成 DNA 的核苷酸、电话网络中的客户或原子核中亚原子粒子的配置。组合学是用于检索信息、设计通信网络、加密交易和 DNA 测序的关键计算机算法的基础。拥有一支在这一尚未进入传统公立学校课程的关键领域拥有专业知识的科学队伍具有经济和战略重要性。 该项目的研究者致力于培训学生并让他们参与研究的各个方面。 这项工作的本质是，引人入胜的开放性问题吸引了本科生和研究生水平的学生，其中许多人在之前与该项目负责人合作的项目中做出了重大贡献。该项目的结果将有助于数学的其他领域，例如有序集和李群的表示论，激光和超导体中玻色子和费米子的统计行为的研究，以及统计数据的可视化。