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FURTHER ARITHMETIC PROPERTIES OF OVERCUBIC PARTITION TRIPLES

Part of: Additive number theory; partitions

Published online by Cambridge University Press:  10 January 2025

MANJIL P. SAIKIA Open the ORCID record for MANJIL P. SAIKIA [Opens in a new window]  and
ABHISHEK SARMA Open the ORCID record for ABHISHEK SARMA [Opens in a new window]
MANJIL P. SAIKIA
Affiliation:
Mathematical and Physical Sciences Division, School of Arts and Sciences, Ahmedabad University, Ahmedabad 380009, Gujarat, India e-mail: manjil@saikia.in
ABHISHEK SARMA*
Affiliation:
Department of Mathematical Sciences, Tezpur University, Napaam, Tezpur 784028, Assam, India
*
e-mail: abhitezu002@gmail.com
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Abstract

We prove several new congruences for the overcubic partition triples function, using both elementary techniques and the theory of modular forms. These extend the recent list of such congruences given by Nayaka, Dharmendra and Kumar [‘Divisibility properties for overcubic partition triples’, Integers 24 (2024), Article no. a80, 9 pages]. We also generalise overcubic partition triples to overcubic partition k-tuples and prove arithmetic properties for these partitions.

Keywords

integer partitionsRamanujan-type congruencesRadu’s algorithmmodular forms

MSC classification

Primary: 11P83: Partitions; congruences and congruential restrictions
Secondary: 11P81: Elementary theory of partitions

Information

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 112 , Issue 2 , October 2025 , pp. 260 - 273
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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 second author was partially supported by an institutional fellowship for doctoral research from Tezpur University, Assam, India.

References

Andrews, G. E., The Theory of Partitions, Cambridge Mathematical Library, 2 (Cambridge University Press, Cambridge, 1998).Google Scholar
Andrews, G. E. and Paule, P., ‘MacMahon’s partition analysis XIV: partitions with $n$ copies of $n$ ’, J. Combin. Theory Ser. A 203(34) (2024), Article no. 105836.Google Scholar
Baruah, N. D. and Das, H., ‘Matching coefficients in the series expansions of certain $q$ -products and their reciprocals’, Ramanujan J. 59(2) (2022), 511–548.CrossRefGoogle Scholar
Chan, H.-C., ‘Ramanujan’s cubic continued fraction and an analog of his “most beautiful identity”’, Int. J. Number Theory 6(3) (2010), 673–680.CrossRefGoogle Scholar
Chan, H.-C., ‘Ramanujan’s cubic continued fraction and Ramanujan type congruences for a certain partition function’, Int. J. Number Theory 6(4) (2010), 819–834.CrossRefGoogle Scholar
Cotron, T., Michaelsen, A., Stamm, E. and Zhu, W., ‘Lacunary eta-quotients modulo powers of primes’, Ramanujan J. 53(2) (2020), 269–284.CrossRefGoogle Scholar
Das, H., Saikia, M. P. and Sarma, A., ‘Arithmetic properties modulo powers of $2$ and $3$ for overpartition $k$ -tuples with odd parts’, Preprint, 2024, arXiv:2409.02929.Google Scholar
Johnson, W. P., An Introduction to $q$ -analysis (American Mathematical Society, Providence, RI, 2020).Google ScholarPubMed
Kim, B., ‘ The overcubic partition function mod 3 ’, in: Ramanujan Rediscovered. Proceedings of a Conference on Elliptic Functions, Partitions, and $q$ -series in Memory of K. Venkatachaliengar, Bangalore, India, June 1–5, 2009 (eds. Andrews, G. E., Askey, R. A., Berndt, B. C., Ramanathan, K. G. and Rankin, R. A.) (Ramanujan Mathematical Society, Mysore, 2010), 157–163.Google Scholar
Kim, B., ‘On partition congruences for overcubic partition pairs’, Commun. Korean Math. Soc. 27(3) (2012), 477–482.CrossRefGoogle Scholar
Nayaka, S. S., Dharmendra, B. N. and Mahesh Kumar, M. C., ‘Divisibility properties for overcubic partition triples’, Integers 24 (2024), Article no. a80, 9 pages.Google Scholar
Radu, C.-S., ‘An algorithmic approach to Ramanujan’s congruences’, Ramanujan J. 20(2) (2009), 215–251.CrossRefGoogle Scholar
Radu, C.-S., ‘An algorithmic approach to Ramanujan–Kolberg identities’, J. Symbolic Comput. 68(1) (2015), 225–253.CrossRefGoogle Scholar
Saikia, M. P., ‘Some missed congruences modulo powers of 2 for $t$ -colored overpartitions’, Bol. Soc. Mat. Mex. (3) 29(1) (2023), Article no. 15, 10 pages.Google Scholar
Saikia, M. P., Sarma, A. and Sellers, J. A., ‘Arithmetic properties for overpartition $k$ –tuples with odd parts modulo powers of 2’, J. Ramanujan Math. Soc., to appear.Google Scholar
Sellers, J. A., ‘Elementary proofs of congruences for the cubic and overcubic partition functions’, Australas. J. Combin. 60 (2014), 191–197.Google Scholar
Sellers, J. A., ‘An elementary proof of a conjecture of Saikia on congruences for $t$ –colored overpartitions’, Bol. Soc. Mat. Mex. (3) 30(2) (2024), Article no. 2.Google Scholar
Smoot, N. A., ‘On the computation of identities relating partition numbers in arithmetic progressions with eta quotients: an implementation of Radu’s algorithm’, J. Symbolic Comput. 104 (2021), 276–311.CrossRefGoogle Scholar
Zhao, H. and Zhong, Z., ‘Ramanujan type congruences for a partition function’, Electron. J. Combin. 18(1) (2011), Article no. 9, 58 pages.CrossRefGoogle Scholar