Abstract
Recently, partitions with fixed or bounded difference between largest and smallest parts have attracted a lot of attention. In this paper, we provide both analytic and combinatorial proofs of the generating function for k-regular partitions with bounded difference kt between largest and smallest parts. Inspired by Franklin’s result, we further find a new proof of the generating function for overpartitions with bounded part differences by using Dousse and Kim’s results on (q, z)-overGaussian polynomials.
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Acknowledgements
The authors would like to thank the referee for valuable comments. This work was supported by the National Natural Science Foundation of China (No. 11871246), the Natural Science Foundation of Fujian Province of China (No. 2019J01328), and the Program for New Century Excellent Talents in Fujian Province University (No. B17160).
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Lin, B.L.S., Zheng, S. k-Regular partitions and overpartitions with bounded part differences. Ramanujan J 56, 685–695 (2021). https://doi.org/10.1007/s11139-020-00311-9
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DOI: https://doi.org/10.1007/s11139-020-00311-9
Keywords
- Combinatorial proof
- Difference between largest and smallest part
- k-Regular partition
- Overpartition
- (q, z)-overGaussian polynomials