The overarching theme of this project is new discoveries in the theory of partitions. These discoveries concern both new objects (e.g. Durfee symbols and k-marked Durfee symbols) and new aspects of venerable problems (e.g. MacMahon's partitions without sequences, Alder's Conjecture, etc.). First the proposers consider symmetry studies of the relevant generating functions for the k-marked Durfee symbols. The next topic is the asymptotics related to partitions with short sequences. Then the proposers study partial fraction methods whose genesis lies in Ramanujan's Lost Notebook (a manuscript studied extensively by the senior PI). In addition the proposers look at questions arising from recent discoveries concerning lecture hall partitions and conclude with further investigations of the long standing Alder Conjecture on which the co-PI has made a major breakthrough.
The proposal continues the training of graduate students, one of whom has started a plausible combinatorial approach to the symmetry study mentioned in the first paragraph. While these topics are based on studies in the theory of partitions (a branch of additive number theory), it is notable that partitions with short sequences have interesting implications in probability theory. Also there has been a spate of recent work revealing fascinating relations between k-marked Durfee symbols and recent developments in the theory of modular forms.
During the tenure of the grant, the PI did research on the theory of partitions and related q-series. The PI published 18 papers and two books during this period. All the work consists revealing more about the inner structure of integer partitions and how information about the related generating functions adds to our knowledge. During this period the "smallest parts function" (which counts the toal number of smallest parts in the partitions of n) was discovered and has been the basis of extensive research by the PI and by many others. The discoveries with regard to parity in partition identities have also been extended by others. In addition, it turned out that prior work on q-orthogonal polynomials has arisen anew in the parity studies. During the grant period, the Co-PI worked on problems arising in integer partitions and re- lated areas and published 15 papers with two papers in preparation. The main achievements includes partial answers to Alder’s conjecture and Bressoud’s conjecture. Both conjectures originated from the famous Rogers-Ramanujan identities concern partitions with part differ- ence conditions. Alder’s conjecture was open for more than 50 years and only a few special cases had been proved until the Co-PI’s work. Recently, it was solved by R. Oliver, M. Jameson, and C. Alfes by asymptotic methods. Bressoud’s conjecture was posed by David Bressoud in 1980 and is still open. The Co-PI in collaboration with Sun Kim has obtained some partial results.
