Abstract
Andrews, Lewis, and Lovejoy introduced the partition function PD(n) as the number of partitions of n with designated summands. They proved that \(PD(3n+2)\) is divisible by 3. Later, Chen, Ji, Jin, and the author introduced the pd-rank for partitions with designated summands and gave a combinatorial interpretation of the congruence. In this paper, we define a crank named the d-crank of partitions with designated summands to provide a new combinatorial interpretation for the congruence of Andrews, Lewis, and Lovejoy for \(PD(3n+2)\). In addition, we derive the generating functions for the d-crank differences of partitions with designated summands modulo 2, 3, 4, and 6. Based on these generating functions, we obtain an equality and some inequalities for d-cranks of partitions with designated summands. We also introduce the d-crank moments weighted by the parity of d-cranks \(\mu _{2k,d}(-1,n)\) and show the positivity of \((-1)^n\mu _{2k,d}(-1,n)\).
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The author would like to thank Robert X. J. Hao and the anonymous referee for helpful suggestions. This work was supported by the National Natural Science Foundation of China (No. 11801139), the Natural Science Foundation of Jiangsu Province of China (No. BK20160855), and the Scientific Research Foundation of Nanjing Institute of Technology (No. YKJ201627).
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Shen, E.Y.Y. A crank of partitions with designated summands. Ramanujan J 57, 785–802 (2022). https://doi.org/10.1007/s11139-021-00462-3
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DOI: https://doi.org/10.1007/s11139-021-00462-3