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

本文拟研究的是涉及组合家族结构和具有本质不同特征的家族之间关系的基本问题。第一部分关注线性不等式约束下的分割和组合。最近的研究表明，该框架为许多分区标识提供了一个公共设置。它与由等级条件、禁忌部位和不同条件决定的家庭产生了惊人的联系。PI试图确定可以用这种方式来描述的家庭，可以捕获的分区统计数据，以及可能获得的新见解。第二部分研究罗杰斯-拉马努金类型的身份。越来越多的人认识到这些恒等式在统计物理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，这些恒等式仍然没有很好地理解组合。PI研究分析组合方面的新工具。项目的第三部分侧重于部分有序集的结构，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关于对称维恩图存在性的开放几何问题。它寻求一种新的方法来解决对称链分解在某些重要偏集上的存在性这一悬而未决的问题。组合学是用来调查、分析和操作结构化数据集的数学：世界范围内的网页，构成DNA的核苷酸，电话网络中的客户，或者原子核中亚原子粒子的结构。组合学是检索信息、设计通信网络、加密交易和测序DNA的关键计算机算法的基础。拥有一支在这一关键领域具有专业知识的科学队伍具有经济和战略重要性，这一领域尚未进入传统的公立学校课程。该项目的研究者致力于培养和参与研究的各个方面的学生。工作的性质,引人注目的开放式问题吸引学生undegraduate和研究生水平,其中许多个人作出了实质性的贡献在之前的项目有了这个P.I.The这个项目的结果将数学有用到其他领域,如命令集和李群表示理论,研究统计行为的激光和超导体的玻色子和费米子和可视化的数据统计。

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.