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A NEW REFINEMENT OF FINE’S PARTITION THEOREM

Part of: Additive number theory; partitions Combinatorics

Published online by Cambridge University Press:  29 March 2021

JIAYU KANG Open the ORCID record for JIAYU KANG [Opens in a new window] ,
RUNQIAO LI Open the ORCID record for RUNQIAO LI [Opens in a new window]  and
ANDREW Y. Z. WANG Open the ORCID record for ANDREW Y. Z. WANG [Opens in a new window]
JIAYU KANG
Affiliation:
School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu611731, P.R. China e-mail: [email protected]
RUNQIAO LI
Affiliation:
School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu611731, P.R. China e-mail: [email protected]
ANDREW Y. Z. WANG*
Affiliation:
School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu611731, P.R. China
*
e-mail: [email protected]
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Abstract

We find a new refinement of Fine’s partition theorem on partitions into distinct parts with the minimum part odd. As a consequence, we obtain two companion partition identities. Both analytic and combinatorial proofs are provided.

Keywords

Fine’s partition theoreminvolutionminimal odd excludant

MSC classification

Primary: 05A17: Partitions of integers
Secondary: 11P81: Elementary theory of partitions
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 104 , Issue 3 , December 2021 , pp. 353 - 361
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Copyright
© 2021 Australian Mathematical Publishing Association Inc.

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References

Andrews, G. E. and Berndt, B. C., Ramanujan’s Lost Notebook, Part II (Springer, New York, 2009).Google Scholar
Andrews, G. E. and Newman, D., ‘Partitions and the minimal excludant’, Ann. Comb. 23 (2019), 249–254.10.1007/s00026-019-00427-wCrossRefGoogle Scholar
Bessenrodt, C. and Pak, I., ‘Partition congruences by involutions’, European J. Combin. 25 (2004), 1139–1149.10.1016/j.ejc.2003.09.018CrossRefGoogle Scholar
Fine, N. J., ‘Some new results on partitions’, Proc. Natl. Acad. Sci. USA 34 (1948), 616–618.10.1073/pnas.34.12.616CrossRefGoogle ScholarPubMed
Fine, N. J., Basic Hypergeometric Series and Applications, Mathematical Surveys and Monographs, 27 (American Mathematical Society, Providence, RI, 1988).10.1090/surv/027CrossRefGoogle Scholar
Kang, J. Y., Li, R. Q. and Wang, A. Y. Z., ‘Partition identities related to the minimal excludant’, Preprint.Google Scholar
Yee, A. J., ‘Bijective proof of a theorem of Fine and related partition identities’, Int. J. Number Theory 5 (2009), 219–228.CrossRefGoogle Scholar