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

摘要奖DMS-0300034的Savage拟议的研究是一个调查的基本问题，涉及组合家庭的结构和家庭之间的关系与本质上不同的characteristics. First部分涉及的划分和组合约束的线性不等式。 最近的研究表明，这个框架为许多分区标识提供了一个公共设置。它产生了令人惊讶的connections与家庭定义的秩条件，禁止部分，和差的条件。 PI试图识别可以以这种方式表征的家族，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认同。这些恒等式在统计物理学和李代数中的重要性越来越受到人们的重视，因此，发现新的Rogers-Ramanujan恒等式的研究激增。 然而，这些身份仍然没有很好地理解组合。 PI研究新的工具来分析组合方面。 本项目的第三部分着重于偏序集的结构，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。 它追求一种新的方法来解决在某些重要的偏序集中对称链分解的存在性这一悬而未决的问题。组合数学是用来调查、分析和处理结构化数据集的数学：万维网的网页、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。组合学是关键计算机算法的基础，用于检索信息、设计通信网络、加密交易和DNA测序。拥有一支在这一关键领域具有专门知识的科学工作队伍具有经济和战略意义，这一领域尚未进入传统的公立学校课程。 该项目的研究人员致力于培训和参与学生在研究的各个方面。 这是工作的性质，令人信服的开放式问题吸引学生在本科生和研究生水平，其中许多人作出了实质性的贡献，在以前的项目与此PI。该项目的结果将是有用的其他领域的数学，如有序集和李群表示论，研究统计行为的玻色子和费米子在激光和超导体，以及统计数据的可视化。

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.