Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
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THE UNIMODALITY OF THE r3-CRANK OF 3-REGULAR OVERPARTITIONS

  • Received : 2023.05.06
  • Accepted : 2023.10.05
  • Published : 2024.05.31
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Abstract

An 𝑙-regular overpartition of n is an overpartition of n with no parts divisible by 𝑙. Recently, the authors introduced a partition statistic called r𝑙-crank of 𝑙-regular overpartitions. Let Mr𝑙(m, n) denote the number of 𝑙-regular overpartitions of n with r𝑙-crank m. In this paper, we investigate the monotonicity property and the unimodality of Mr3(m, n). We prove that Mr3(m, n) ≥ Mr3(m, n - 1) for any integers m and n ≥ 6 and the sequence {Mr3(m, n)}|m|≤n is unimodal for all n ≥ 14.

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Acknowledgement

We are grateful to the referee for helpful suggestions.

References

  1. G. E. Andrews, Singular overpartitions, Int. J. Number Theory 11 (2015), no. 5, 1523-1533. https://doi.org/10.1142/S1793042115400059
  2. G. E. Andrews and F. G. Garvan, Dyson's crank of a partition, Bull. Amer. Math. Soc. (N.S.) 18 (1988), no. 2, 167-171. https://doi.org/10.1090/S0273-0979-1988-15637-6
  3. S. Corteel and J. Lovejoy, Overpartitions, Trans. Amer. Math. Soc. 356 (2004), no. 4, 1623-1635. https://doi.org/10.1090/S0002-9947-03-03328-2
  4. R. X. J. Hao and E. Y. Y. Shen, On the number of 𝑙-regular overpartitions, Int. J. Number Theory 17 (2021), no. 9, 2153-2173. https://doi.org/10.1142/S1793042121500810
  5. K. Q. Ji and W. J. T. Zang, Unimodality of the Andrews-Garvan-Dyson cranks of partitions, Adv. Math. 393 (2021), Paper No. 108053, 54 pp. https://doi.org/10.1016/j.aim.2021.108053
  6. J. Lovejoy, Gordon's theorem for overpartitions, J. Combin. Theory Ser. A 103 (2003), no. 2, 393-401. https://doi.org/10.1016/S0097-3165(03)00116-X
  7. S. Ramanujan, Some properties of p(n), the number of partitons of n, Proc. Cambridge Philos. Soc. 19 (1919), 207-210.
  8. E. Y. Y. Shen, Arithmetic properties of 𝑙-regular overpartitions, Int. J. Number Theory 12 (2016), no. 3, 841-852. https://doi.org/10.1142/S1793042116500548