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

本文研究的是关于组合族的结构以及具有本质不同特征的族之间的关系的基本问题。第一部分是关于线性不等式约束的划分和合成。最近的研究表明，该框架为许多分区标识提供了公共设置。它与由等级条件、禁止部分和不同条件定义的家庭产生了令人惊讶的联系。PI试图确定可以用这种方式表征的族、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了Rogers-Ramanujan类型的身份。人们越来越多地认识到这些恒等式在统计物理学和Lie代数中的重要性，因此，发现罗杰斯-拉马努扬类型的新恒等式的研究激增。尽管如此，这些身份仍然没有得到很好的组合理解。PI研究了新的工具来分析组合方面。该项目的第三部分集中在偏序集中的结构，具体地说，对称链分解。这推广了PI和他的同事最近的工作，他们使用对称链分解来解决关于对称维恩图的存在的开放几何问题。它探索了一种新的方法来解决某些重要位置上存在对称链分解这一悬而未决的问题。组合学是用于调查、分析和操作结构化数据集的数学，这些数据集包括万维网的页面、形成DNA的核苷酸、电话网络中的客户或原子核中的亚原子粒子的构型。组合学是检索信息、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。拥有一支在这一关键领域拥有专门知识的科学队伍具有重要的经济和战略意义，这一领域尚未进入传统的公立学校课程。这个项目的调查者致力于培训学生并让他们参与到研究的各个方面。这项工作的本质是，引人注目的开放问题吸引了本科生和研究生水平的学生，他们中的许多人在这个P.I.之前的项目中做出了实质性的贡献。这个项目的结果将对其他数学领域有用，如有序集和李群表示理论，研究激光和超导体中玻色子和费米子的统计行为，以及统计学中数据的可视化。

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.