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Annals of Combinatorics

, Volume 22, Issue 4, pp 803–818 | Cite as

On the Crank Function of Cubic Partition Pairs

  • Byungchan Kim
  • Pee Choon Toh
Article
  • 33 Downloads

Abstract

We study a crank function M(mn) for cubic partition pairs. We show that the function M(mn) explains a cubic partition pair congruence and we also obtain various arithmetic properties regarding M(mn). In particular, using the \(\Theta \)-operator, we confirm a conjecture on the sign pattern of c(n), the number of cubic partition pairs of n, weighted by the parity of the crank.

Keywords

Partitions Partition crank Cubic partitions Congruences Modular forms 

Mathematics Subject Classification

05A17 11P83 

Notes

Acknowledgements

The authors are grateful to anonymous referees for their careful readings and helpful comments on an earlier version of this paper.

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Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.School of Liberal ArtsSeoul National University of Science and TechnologyNowonguRepublic of Korea
  2. 2.Mathematics and Mathematics Education, National Institute of EducationNanyang Technological UniversitySingaporeSingapore

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