The proposed research is an investigation of fundamental questions involving the structure of combinatorial families and relationships between 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 connections with families defined by rank conditions, by forbidden parts, and by difference conditions. The PI seeks to identify the families that can be characterized in this way, the partition statistics that can be captured, and the new insight that might be gained. The second part of the research studies identities of the Rogers-Ramanujan type. There has been a growing recognition of the importance of these identities in statistical physics and Lie 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 chain decompositions. 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 decomposition in certain important posets.
Combinatorics is the mathematics used to investigate, analyze, and manipulate structured data sets: the pages of the world-wide web, the nucleotides forming DNA, the customers in a telephone network, or the configuration of subatomic particles in the nucleus of atoms. Combinatorics underlies critical computer 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 whom have 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.
