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Construction of Evidently Positive Series and an Alternative Construction for A Family of Partition Generating Functions Due to Kanade and Russell

Abstract

We give an alternative construction for a family of partition generating functions due to Kanade and Russell. In our alternative construction, we use ordinary partitions instead of jagged partitions. We also present new generating functions which are evidently positive series for partitions due to Kanade and Russell. To obtain those generating functions, we first construct an evidently positive series for a key infinite product. In that construction, a series of combinatorial moves is used to decompose an arbitrary partition into a base partition together with some auxiliary partitions that bijectively record the moves.

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References

  1. K. Alladi, Partition identities involving gaps and weights, Transactions of the American Mathematical Society, 349(12), 5001–5019 (1997).

    MathSciNet  Article  Google Scholar 

  2. G. E. Andrews, Number Theory, Dover Publications, New York (1994).

  3. G. E. Andrews, The Theory of Partitions, Cambridge University Press, Cambridge (1984).

  4. K. Bringmann, C. Jennings-Shaffer and K. Mahlburg, Proofs and reductions of various conjectured partition identities of Kanade and Russell. Journal fur die Reine und Angewandte Mathematik (Crelle’s Journal) 766, 109-135 (2020).

    MathSciNet  Article  Google Scholar 

  5. L. Euler, Introduction to Analysis of the Infinite, The Electronic Journal of Combinatorics, transl. by J. Blanton, Springer, New York (1988).

  6. S. Kanade, M. C. Russell, Staircases to analytic sum-sides for many new integer partition identities of Rogers-Ramanujan type, The Electronic Journal of Combinatorics, 26(1), 1-6 (2019).

    MathSciNet  Article  Google Scholar 

  7. K. Kurşungöz, Andrews-Gordon type series for Capparelli’s and Göllnitz-Gordon identities, Journal of Combinatorial Theory, Series A, 165, 117-138 (2019).

    MathSciNet  Article  Google Scholar 

  8. K. Kurşungöz, Andrews-Gordon type series for Kanade-Russell conjectures, Annals of Combinatorics, 23, 835-888 (2019).

    MathSciNet  Article  Google Scholar 

  9. K. Kurşungöz, Andrews-Gordon type series for Schur’s partition identity, Discrete Mathematics, 344(11), 112563 (2021).

    MathSciNet  Article  Google Scholar 

  10. H. Rosengren, Proofs of some partition identities conjectured by Kanade and Russell, The Ramanujan Journal, 1-23 (2021).

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Acknowledgements

The majority of the results in this paper are from the second author’s Ph.D. dissertation. The first author acknowledges Bilim Akademisi BAGEP support with gratitute. The authors thank the referees for the constructive comments, recommendations, and for pointing out the reference [1]; which definitely helped to improve the readability of the paper.

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Correspondence to Halme Ömrüuzun Seyrek.

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Communicated by Sylvie Corteel.

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Kurşungöz, K., Ömrüuzun Seyrek, H. Construction of Evidently Positive Series and an Alternative Construction for A Family of Partition Generating Functions Due to Kanade and Russell. Ann. Comb. (2022). https://doi.org/10.1007/s00026-022-00597-0

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  • DOI: https://doi.org/10.1007/s00026-022-00597-0

Keywords

  • Integer partition
  • Partition generating function
  • Evidently positive generating functions
  • Rogers-Ramanujan type partition identities