Number Theory and Combinatorics

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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.