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ARITHMETIC PROPERTIES OF 3-REGULAR PARTITIONS IN THREE COLOURS

Part of: Combinatorics Additive number theory; partitions

Published online by Cambridge University Press:  07 June 2021

ROBSON DA SILVA Open the ORCID record for ROBSON DA SILVA [Opens in a new window]  and
JAMES A. SELLERS
ROBSON DA SILVA*
Affiliation:
Departamento de Ciência e Tecnologia, Universidade Federal de São Paulo, Av. Cesare M. G. Lattes, 1201, São José dos Campos, SP, 12247-014, Brazil
JAMES A. SELLERS
Affiliation:
Department of Mathematics and Statistics, University of Minnesota Duluth, Duluth, MN55812, USA e-mail: jsellers@d.umn.edu
*
e-mail: silva.robson@unifesp.br
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Abstract

Gireesh and Mahadeva Naika [‘On 3-regular partitions in 3-colors’, Indian J. Pure Appl. Math.50 (2019), 137–148] proved an infinite family of congruences modulo powers of 3 for the function $p_{\{3,3\}}(n)$ , the number of 3-regular partitions in three colours. In this paper, using elementary generating function manipulations and classical techniques, we significantly extend the list of proven arithmetic properties satisfied by $p_{\{3,3\}}(n).$

Keywords

partition3-regularthree colourscongruence

MSC classification

Primary: 11P83: Partitions; congruences and congruential restrictions
Secondary: 05A17: Partitions of integers
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 104 , Issue 3 , December 2021 , pp. 415 - 423
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Copyright
© 2021 Australian Mathematical Publishing Association Inc.

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Footnotes

The first author was supported by the São Paulo Research Foundation (FAPESP) (grant no. 2019/14796-8).

References

Berndt, B. C., Ramanujan’s Notebooks, Part III (Springer, New York, 1991).CrossRefGoogle Scholar
Berndt, B. C., Number Theory in the Spirit of Ramanujan (American Mathematical Society, Providence, RI, 2006).CrossRefGoogle Scholar
Boll, E. and Penniston, D., ‘The 7-regular and 13-regular partition functions modulo 3’, Bull. Aust. Math. Soc. 93 (2016), 410–419.10.1017/S0004972715001434CrossRefGoogle Scholar
Chern, S. and Hao, L.-J., ‘Congruences for partition functions related to mock theta functions’, Ramanujan J. 48 (2019), 369–384.10.1007/s11139-017-9979-1CrossRefGoogle Scholar
da Silva, R. and Sellers, J. A., ‘New congruences for 3-regular partitions with designated summands’, Integers 20A (2020), #A6.Google Scholar
da Silva, R. and Sellers, J. A., ‘Infinitely many congruences for $k$ -regular partitions with designated summands’, Bull. Braz. Math. Soc. 51 (2020), 357–370.CrossRefGoogle Scholar
Dou, D. Q., Guo, L. and Lin, B. L. S., ‘Parity results for 13-regular partitions and broken 6-diamond partitions’, Ramanujan J. 44 (2017), 521–530.CrossRefGoogle Scholar
Gireesh, D. S. and Mahadeva Naika, M. S., ‘On 3-regular partitions in 3-colors’, Indian J. Pure Appl. Math. 50 (2019), 137–148.CrossRefGoogle Scholar
Hirschhorn, M. D., The Power of $q$ : A Personal Journey, Developments in Mathematics, 49 (Springer, New York, 2017).Google Scholar
Hirschhorn, M. D., ‘Partitions in 3 colours’, Ramanujan J. 45 (2018), 399–411.CrossRefGoogle Scholar
Hirschhorn, M. D. and Sellers, J. A., ‘Elementary proofs of parity results for 5-regular partitions’, Bull. Aust. Math. Soc. 81(1) (2010), 58–63.10.1017/S0004972709000525CrossRefGoogle Scholar
Keith, W. J., ‘Congruences for $m$ -regular partitions modulo 4’, Integers 15A (2015), #A11.Google Scholar
Wang, L., ‘Arithmetic properties of $\left(k,\ell \right)$ -regular bipartitions’, Bull. Aust. Math. Soc. 95(3) (2017), 353–364.CrossRefGoogle Scholar
Wang, L., ‘Congruences for 5-regular partitions modulo powers of 5’, Ramanujan J. 44(2) (2017), 343–358.10.1007/s11139-015-9767-8CrossRefGoogle Scholar