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On a Conjecture of Hanna Connecting Distinct Part and Complete Partitions

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Complete partitions are a generalization of MacMahon’s perfect partitions; we further generalize these by defining k-step partitions. A matrix equation shows an unexpected connection between k-step partitions and distinct part partitions. We provide two proofs of the corresponding theorem, one using generating functions and one combinatorial. The algebraic proof relies on a generalization of a conjecture made by Paul Hanna in 2012.

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Correspondence to Brian Hopkins.

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Andrews, G.E., Beck, G. & Hopkins, B. On a Conjecture of Hanna Connecting Distinct Part and Complete Partitions. Ann. Comb. (2020). https://doi.org/10.1007/s00026-019-00476-1

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  • Integer partitions
  • Distinct part partitions
  • Complete partitions

Mathematics Subject Classification

  • 05A17
  • 11P84