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INFINITE FAMILIES OF CONGRUENCES MODULO 3 AND 9 FOR BIPARTITIONS WITH 3-CORES

Part of: Additive number theory; partitions Combinatorics

Published online by Cambridge University Press:  27 August 2014

OLIVIA X. M. YAO
OLIVIA X. M. YAO*
Affiliation:
Department of Mathematics, Jiangsu University, Jiangsu, Zhenjiang 212013, PR China email yaoxiangmei@163.com
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Abstract

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Let $A_{3}(n)$ denote the number of bipartitions of $n$ with 3-cores. Recently, Lin [‘Some results on bipartitions with 3-core’, J. Number Theory139 (2014), 44–52] established some congruences modulo 4, 5, 7 and 8 for $A_{3}(n)$. In this paper, we prove several infinite families of congruences modulo 3 and 9 for $A_{3}(n)$ by employing two identities due to Ramanujan.

Keywords

bipartitionst-corescongruences

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 91 , Issue 1 , February 2015 , pp. 47 - 52
Copyright
Copyright © 2014 Australian Mathematical Publishing Association Inc. 

References

Atkin, A. O. L., ‘Ramanujan congruences for p −k(n)’, Canad. J. Math. 20 (1968), 67–78; Canad. J. Math. 21 (1968), 256 (corrigendum).CrossRefGoogle Scholar
Baruah, N. D. and Nath, K., ‘Some results on 3-cores’, Proc. Amer. Math. Soc. 142 (2014), 441–448.Google Scholar
Berndt, B. C., Ramanujan’s Notebooks, Part III (Springer, New York, 1991).Google Scholar
Bringmann, K. and Lovejoy, J., ‘Rank and congruences for overpartition pairs’, Int. J. Number Theory 4 (2008), 303–322.CrossRefGoogle Scholar
Chan, S. H. and Mao, R., ‘Pairs of partitions without repeated odd parts’, J. Math. Anal. Appl. 394 (2012), 408–415.Google Scholar
Chen, S., ‘Arithmetical properties of the number of t-core partitions’, Ramanujan J. 18 (2009), 103–112.CrossRefGoogle Scholar
Chen, S. C., ‘Congruences for t-core partition functions’, J. Number Theory 133 (2013), 4036–4046.Google Scholar
Gandhi, J. M., ‘Congruences for p r(n) and Ramanujan’s 𝜏-function’, Amer. Math. Monthly 70 (1963), 265–274.Google Scholar
Garvan, F., ‘Some congruences for partitions that are p-cores’, Proc. Lond. Math. Soc. (3) 66 (1993), 449–478.Google Scholar
Garvan, F., Kim, D. and Stanton, D., ‘Cranks and t-cores’, Invent. Math. 101 (1990), 1–17.CrossRefGoogle Scholar
Granville, A. and Ono, K., ‘Defect zero p-blocks for infinite simple groups’, Trans. Amer. Math. Soc. 348 (1996), 331–347.Google Scholar
Hirschhorn, M. D. and Sellers, J. A., ‘Elementary proofs of various facts about 3-cores’, Bull. Aust. Math. Soc. 79 (2009), 507–512.Google Scholar
Lin, B. L. S., ‘Arithmetic properties of bipartitions with even parts distinct’, Ramanujan J. 33 (2014), 269–279.Google Scholar
Lin, B. L. S., ‘Some results on bipartitions with 3-core’, J. Number Theory 139 (2014), 44–52.CrossRefGoogle Scholar
Lin, B. L. S., ‘Arithmetic of the 7-regular bipartition function modulo 3’, Ramanujan J., to appear; doi:10.1007/s11139-013-9542-7.Google Scholar
Ramanathan, K. G., ‘Identities and congruences of the Ramanujan type’, Canad. J. Math. 2 (1950), 168–178.Google Scholar
Radu, S. and Sellers, J. A., ‘Parity results for broken k-diamond partitions and (2k + 1)-cores’, Acta Arith. 146 (2011), 43–52.Google Scholar
Toh, P. C., ‘Ramanujan type identities and congruences for partition pairs’, Discrete Math. 312 (2012), 1244–1250.Google Scholar