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COMBINATORIAL PROOFS FOR TWO-COLOUR PARTITIONS

Part of: Combinatorics Additive number theory; partitions Diophantine equations

Published online by Cambridge University Press:  11 April 2025

DANDAN CHEN Open the ORCID record for DANDAN CHEN [Opens in a new window]  and
ZIYIN ZOU Open the ORCID record for ZIYIN ZOU [Opens in a new window]
DANDAN CHEN*
Affiliation:
Department of Mathematics, Shanghai University, Shanghai, PR China and Newtouch Center for Mathematics of Shanghai University, Shanghai University, Shanghai, PR China
ZIYIN ZOU
Affiliation:
Department of Mathematics, Shanghai University, Shanghai, PR China e-mail: [email protected]
*
e-mail: [email protected]
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Abstract

Andrews and El Bachraoui [‘On two-colour partitions with odd smallest part’, Preprint, arXiv:2410.14190] explored many integer partitions in two colours, some of which are generated by the mock theta functions of third order of Ramanujan and Watson. They also posed questions regarding combinatorial proofs for these results. In this paper, we establish bijections to provide a combinatorial proof of one of these results and a companion result. We give analytic proofs of further companion results.

Keywords

integer partitionsq-series

MSC classification

Primary: 11P81: Elementary theory of partitions
Secondary: 05A17: Partitions of integers 11D09: Quadratic and bilinear equations
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , First View , pp. 1 - 5
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Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.

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Footnotes

The first author was supported by the National Key R&D Program of China (Grant No. 2024YFA1014500) and National Natural Science Foundation of China (Grant No. 12201387).

References

Andrews, G. E., The Theory of Partitions (Cambridge University Press, Cambridge, 1998).Google Scholar
Andrews, G. E. and El Bachraoui, M., ‘On two-color partitions with odd smallest part’, Preprint, 2024, arXiv:2410.14190.Google Scholar
Corteel, S. and Lovejoy, J., ‘Overpartitions’, Trans. Amer. Math. Soc. 356 (2004), 1623–1635.CrossRefGoogle Scholar
Hirschhorn, M. D. and Sellers, J. A., ‘Arithmetic properties of overpartitions into odd parts’, Ann. Comb. 10 (2006), 353–367.CrossRefGoogle Scholar