Abstract
Andrews, Lewis, and Lovejoy introduced the partition function PD(n) as the number of partitions of n with designated summands. A bipartition of n is an ordered pair of partitions \((\pi _1, \pi _2)\) with the sum of all of the parts being n. In this paper, we introduce a generalized crank named the pd-crank for bipartitions with designated summands and give some inequalities for the pd-crank of bipartitions with designated summands modulo 2 and 3. We also define the pd-crank moments weighted by the parity of pd-cranks \(\mu _{2k,bd}(-1,n)\) and show the positivity of \((-1)^n\mu _{2k,bd}(-1,n)\). Let \(M_{bd}(m,n)\) denote the number of bipartitions of n with designated summands with pd-crank m. We prove a monotonicity property of pd-cranks of bipartitions with designated summands and find that the sequence \(\{M_{bd}(m,n)\}_{|m|\le n}\) is unimodal for \(n\not = 1,5,7\).
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The authors would like to thank anonymous referee for his/her valuable comments.
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Robert X. J. Hao was supported by the Scientific Research Foundation of Nanjing Institute of Technology (No. YKJ201627). Erin Y. Y. Shen was supported by the National Natural Science Foundation of China (No. 11801139) and the Natural Science Foundation of Jiangsu Province of China (No. BK20160855)
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Hao, R.X.J., Shen, E.Y.Y. A crank for bipartitions with designated summands. Ramanujan J 56, 785–802 (2021). https://doi.org/10.1007/s11139-021-00492-x
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DOI: https://doi.org/10.1007/s11139-021-00492-x