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Surprising connection between distinct and complete partitions

Brian Hopkins
Mathematics Department, Saint Peter's University, USA

16:00 - 17:00 Wednesday 23th January 2019, Room A418, MUIC


I will show the way from mysterious matrix identities involving integer partitions to both generating function and combinatorial proofs. Partitions with distinct parts were part of history's first notable work on partitions, done by Euler. MacMahon considered perfect partitions of n, where 1, 2, …, n are each the sum of a unique subpartition. Park relaxed the uniqueness criterion to define complete partitions. We have generalized complete partitions with a notion of tightness. Another ingredient is the program of Schneider generalizing classic number theory to partitions, including the Möbius function. All of this contributes to unexpected connections seen in matrices. The proofs explaining this are elementary, using generating functions and combinatorial bijections. This is joint work with George Andrews and George Beck.


Last modified: 2019/01/30 15:26