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GENERALISATION OF KEITH’S CONJECTURE ON 9-REGULAR PARTITIONS AND 3-CORES

Part of: Combinatorics Additive number theory; partitions

Published online by Cambridge University Press:  20 May 2014

BERNARD L. S. LIN  and
ANDREW Y. Z. WANG
BERNARD L. S. LIN
Affiliation:
School of Sciences, Jimei University, Xiamen 361021, PR China email [email protected]
ANDREW Y. Z. WANG*
Affiliation:
School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu 611731, PR China email [email protected]
*
✉[email protected]
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Abstract

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Recently, Keith used the theory of modular forms to study 9-regular partitions modulo 2 and 3. He obtained one infinite family of congruences modulo 3, and meanwhile proposed an analogous conjecture. In this note, we show that 9-regular partitions and 3-cores satisfy the same congruences modulo 3. Thus, we first derive several results on 3-cores, and then generalise Keith’s conjecture and get a stronger result, which implies that all of Keith’s results on congruences modulo 3 are consequences of our result.

Keywords

regular partitionscongruences3-cores

MSC classification

Secondary: 05A17: Partitions of integers 11P83: Partitions; congruences and congruential restrictions
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 90 , Issue 2 , October 2014 , pp. 204 - 212
Copyright
Copyright © 2014 Australian Mathematical Publishing Association Inc. 

References

Ahlgren, S. and Lovejoy, J., ‘The arithmetic of partitions into distinct parts’, Mathematika 48 (2001), 203–211.Google Scholar
Andrews, G. E., Hirschhorn, M. D. and Sellers, J. A., ‘Arithmetic properties of partitions with even parts distinct’, Ramanujan J. 23 (2010), 169–181.CrossRefGoogle Scholar
Baruah, N. D. and Nath, K., ‘Two quotients of theta functions and arithmetic identities for 3-cores’, in: The Legacy of Srinivasa Ramanuajan, RMS Lecture Notes Series 20 (eds. Berndt, B. C. and Prasad, D.) (Ramanujan Mathematical Society, 2013), 99–110.Google Scholar
Baruah, N. D. and Nath, K., ‘Some results on 3-cores’, Proc. Amer. Math. Soc. 142 (2014), 441–448.CrossRefGoogle Scholar
Calkin, N., Drake, N., James, K., Law, S., Lee, P., Penniston, D. and Radder, J., ‘Divisibility properties of the 5-regular and 13-regular partition functions’, Integers 8 (2008), A60.Google Scholar
Chen, S. C., ‘On the number of partitions with distinct even parts’, Discrete Math. 311 (2011), 940–943.CrossRefGoogle Scholar
Cui, S. P. and Gu, N. S. S., ‘Arithmetic properties of the ℓ-regular partitions’, Adv. in Appl. Math. 51 (2013), 507–523.Google Scholar
Dandurand, B. and Penniston, D., ‘ℓ-divisibility of ℓ-regular partition functions’, Ramanujan J. 19 (2009), 63–70.CrossRefGoogle Scholar
Furcy, D. and Penniston, D., ‘Congruences for ℓ-regular partition functions modulo 3’, Ramanujan J. 27 (2012), 101–108.CrossRefGoogle Scholar
Gordon, B. and Ono, K., ‘Divisibility of certain partition functions by powers of primes’, Ramanujan J. 1 (1997), 25–34.Google Scholar
Granville, A. and Ono, K., ‘Defect zero p-blocks for finite 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
Hirschhorn, M. D. and Sellers, J. A., ‘Elementary proofs of parity results for 5-regular partitions’, Bull. Aust. Math. Soc. 81 (2010), 58–63.Google Scholar
Keith, W. J., ‘Congruences for 9-regular partitions modulo 3’, Ramanujan J. , to appear.Google Scholar
Lin, B. L. S., ‘Arithmetic of the 7-regular bipartition function modulo 3’, Ramanujan J. , to appear.Google Scholar
Lovejoy, J., ‘The divisibility and distribution of partitions into distinct parts’, Adv. Math. 158 (2001), 253–263.Google Scholar
Lovejoy, J., ‘The number of partitions into distinct parts modulo powers of 5’, Bull. Lond. Math. Soc. 35 (2003), 41–46.Google Scholar
Lovejoy, J. and Penniston, D., ‘3-regular partitions and a modular K3 surface’, Contemp. Math. 291 (2001), 177–182.Google Scholar
Ono, K. and Penniston, D., ‘The 2-adic behavior of the number of partitions into distinct parts’, J. Combin. Theory Ser. A 92 (2000), 138–157.Google Scholar
Penniston, D., ‘The p a-regular partition function modulo p j’, J. Number Theory 94 (2002), 320–325.CrossRefGoogle Scholar
Penniston, D., ‘Arithmetic of ℓ-regular partition functions’, Int. J. Number Theory 4 (2008), 295–302.CrossRefGoogle Scholar
Webb, J. J., ‘Arithmetic of the 13-regular partition function modulo 3’, Ramanujan J. 25 (2011), 49–56.Google Scholar
Xia, E. X. W. and Yao, O. X. M., ‘A proof of Keith’s conjecture for 9-regular partitions modulo 3’, Int. J. Number Theory 10 (2014), 669–674.Google Scholar
Xia, E. X. W. and Yao, O. X. M., ‘Parity results for 9-regular partitions’, Ramanujan J. , to appear.Google Scholar