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On Conjectures Concerning the Smallest Part and Missing Parts of Integer Partitions

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Abstract

For positive integers \(L \ge 3\) and s, Berkovich and Uncu (Ann Comb 23:263–284, 2019) conjectured an inequality between the sizes of two closely related sets of partitions whose parts lie in the interval \(\{s, \ldots , L+s\}\). Further restrictions are placed on the sets by specifying impermissible parts as well as a minimum part. The authors proved their conjecture for the cases \(s=1\) and \(s=2\). In the present article, we prove their conjecture for general s by proving a stronger theorem. We also prove other related conjectures found in the same paper.

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Notes

  1. For recent progress on both Conjectures 1 and 2, see Sect. 1.1.

  2. See Footnote 1, Page 2.

References

  1. G.E. Andrews, M. Beck, and N. Robbins. Partitions with fixed differences between largest and smallest parts. Proc. Amer. Math. Soc., 143(10):4283–4289, 2015.

  2. J. L. Ramírez Alfonsín. The Diophantine Frobenius problem, volume 30 of Oxford Lecture Series in Mathematics and its Applications. Oxford University Press, Oxford, 2005.

  3. D. Binner. The number of solutions to \(ax+by+cz=n\) and its relation to quadratic residues. Journal of Integer Sequences, 23(20.6.5), 2020.

  4. A. Berkovich and A. K. Uncu. New weighted partition theorems with the emphasis on the smallest part of partitions. In Analytic number theory, modular forms and \(q\)-hypergeometric series, volume 221 of Springer Proc. Math. Stat., pages 69–94. Springer, Cham, 2017.

  5. A. Berkovich and A. K. Uncu. Some elementary partition inequalities and their implications. Ann. Comb., 23:263–284, 2019.

  6. R. Chapman. Partitions with bounded differences between largest and smallest parts. Australas. J. Combin., 64:376–378, 2016.

  7. J. J. Sylvester. On Subinvariants, i.e. Semi-invariants to Binary Quantics of an Unlimited Order. Amer. J. Math., 5(1):79–136, 1882.

  8. W. J. T. Zang and J. Zeng. Gap between the largest and smallest parts of partitions and Berkovich and Uncu’s conjectures, 2020, arXiv:2004.12871 [math.CO].

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Acknowledgements

We thank the reviewers for their encouraging comments and their many helpful suggestions.

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Authors and Affiliations

  1. Department of Mathematics, Simon Fraser University, Burnaby, BC, V5A 1S6, Canada

    Damanvir Singh Binner & Amarpreet Rattan

Authors
  1. Damanvir Singh Binner
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  2. Amarpreet Rattan
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Correspondence to Damanvir Singh Binner.

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Communicated by Matjaz Konvalinka

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Binner, D.S., Rattan, A. On Conjectures Concerning the Smallest Part and Missing Parts of Integer Partitions. Ann. Comb. 25, 697–728 (2021). https://doi.org/10.1007/s00026-021-00528-5

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  • DOI: https://doi.org/10.1007/s00026-021-00528-5

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