On a Conjecture of Hanna Connecting Distinct Part and Complete Partitions

  • George E. Andrews
  • George Beck
  • Brian HopkinsEmail author


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.


Integer partitions Distinct part partitions Complete partitions 

Mathematics Subject Classification

05A17 11P84 



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Copyright information

© Springer Nature Switzerland AG 2020

Authors and Affiliations

  • George E. Andrews
    • 1
  • George Beck
    • 2
  • Brian Hopkins
    • 3
    Email author
  1. 1.The Pennsylvania State UniversityUniversity ParkUSA
  2. 2.Wolfram Research, Inc.ChampaignUSA
  3. 3.Saint Peter’s UniversityJersey CityUSA

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