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GENERALISATION OF A RESULT ON DISTINCT PARTITIONS WITH BOUNDED PART DIFFERENCES

  • RUNQIAO LI (a1), BERNARD L. S. LIN (a2) and ANDREW Y. Z. WANG (a3)

Abstract

We generalise a result of Chern [‘A curious identity and its applications to partitions with bounded part differences’, New Zealand J. Math. 47 (2017), 23–26] on distinct partitions with bounded difference between largest and smallest parts. The generalisation is proved both analytically and bijectively.

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This work was supported by the National Natural Science Foundation of China (Nos. 11401080 and 11871246).

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[1] Andrews, G. E., The Theory of Partitions (Cambridge University Press, New York, 1998).
[2] Andrews, G. E., Beck, M. and Robbins, N., ‘Partitions with fixed differences between largest and smallest parts’, Proc. Amer. Math. Soc. 143 (2015), 42834289.
[3] Breuer, F. and Kronholm, B., ‘A polyhedral model of partitions with bounded differences and a bijective proof of a theorem of Andrews, Beck, and Robbins’, Res. Number Theory 2 (2016), Article 2, 15 pages.
[4] Chapman, R., ‘Partitions with bounded differences between largest and smallest parts’, Australas. J. Combin. 64 (2016), 376378.
[5] Chern, S., ‘A curious identity and its applications to partitions with bounded part differences’, New Zealand J. Math. 47 (2017), 2326.
[6] Chern, S., ‘An overpartition analogue of partitions with bounded differences between largest and smallest parts’, Discrete Math. 340 (2017), 28342839.
[7] Chern, S. and Yee, A. J., ‘Overpartitions with bounded part differences’, Eur. J. Combin. 70 (2018), 317324.
[8] Lin, B. L. S., ‘ $k$ -regular partitions with bounded differences between largest and smallest parts’, Preprint.
[9] Lin, B. L. S., ‘Refinements of the results on partitions and overpartitions with bounded part differences’, Preprint.
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