Abstract
Andrews, Lewis, and Lovejoy introduced partitions with designated summands. Let PD(n) enumerate partitions with designated summands of n. Andrews et al. gave a Ramanujan-type congruence mod 3 satisfied by PD(n). Recently, Shen defined the d-crank on partitions with designated summands to provide a combinatorial interpretation for this congruence. Shen also derived some equalities and inequalities on d-cranks. In this paper, we refine some equalities and inequalities given by Shen. Meanwhile, we establish a refinement of the Ramanujan-type congruence of Andrews et al.
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Acknowledgements
The authors would like to thank S. P. Cui, Erin Y. Y. Shen, and the anonymous referee for their helpful comments and suggestions that have improved this paper to a great extent. This work was supported by the National Natural Science Foundation of China (No. 12101307) and the Qing Lan Project of JiangSu Province.
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Communicated by Emrah Kilic.
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Hao, R.X.J., Zhang, Y. Refinements of Equalities and Inequalities on d-Cranks for Partitions with Designated Summands. Bull. Malays. Math. Sci. Soc. 45, 1987–2002 (2022). https://doi.org/10.1007/s40840-022-01345-6
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DOI: https://doi.org/10.1007/s40840-022-01345-6