Hostname: page-component-8448b6f56d-jr42d Total loading time: 0 Render date: 2024-04-24T07:33:20.841Z Has data issue: false hasContentIssue false
  • English
  • Français

CONGRUENCES MODULO 2 FOR CERTAIN PARTITION FUNCTIONS

Part of: Combinatorics Additive number theory; partitions

Published online by Cambridge University Press:  11 November 2015

M. S. MAHADEVA NAIKA ,
B. HEMANTHKUMAR  and
H. S. SUMANTH BHARADWAJ
M. S. MAHADEVA NAIKA*
Affiliation:
Department of Mathematics, Bangalore University, Central College Campus, Bengaluru-560 001, Karnataka, India email msmnaika@rediffmail.com
B. HEMANTHKUMAR
Affiliation:
Department of Mathematics, Bangalore University, Central College Campus, Bengaluru-560 001, Karnataka, India email hemanthkumarb.30@gmail.com
H. S. SUMANTH BHARADWAJ
Affiliation:
Department of Mathematics, Bangalore University, Central College Campus, Bengaluru-560 001, Karnataka, India email sumanthbharadwaj@gmail.com
*
✉msmnaika@rediffmail.com
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Let $b_{3,5}(n)$ denote the number of partitions of $n$ into parts that are not multiples of 3 or 5. We establish several infinite families of congruences modulo 2 for $b_{3,5}(n)$. In the process, we also prove numerous parity results for broken 7-diamond partitions.

Keywords

congruencepartitiontheta function

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 93 , Issue 3 , June 2016 , pp. 400 - 409
Copyright
© 2015 Australian Mathematical Publishing Association Inc. 

References

Ahmed, Z. and Baruah, N. D., ‘Parity results for broken 5-diamond, 7-diamond and 11-diamond partitions’, Int. J. Number Theory 11 (2015), 527–542.CrossRefGoogle Scholar
Andrews, G. E. and Paule, P., ‘MacMahon’s partition analysis XI: broken diamonds and modular forms’, Acta Arith. 126 (2007), 281–294.CrossRefGoogle Scholar
Baruah, N. D. and Berndt, B. C., ‘Partition identities and Ramanujan’s modular equations’, J. Combin. Theory Ser. A 114 (2007), 1024–1045.CrossRefGoogle Scholar
Berndt, B. C., Ramanujan’s Notebooks, Part III (Springer, New York, 1991).Google 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
Cui, S. P. and Gu, N. S. S., ‘Arithmetic properties of l-regular partitions’, Adv. Appl. Math. 51 (2013), 507–523.CrossRefGoogle Scholar
Furcy, D. and Penniston, D., ‘Congruences for ℓ-regular partition functions modulo 3’, Ramanujan J. 27 (2012), 101–108.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. 35 (2014), 157–164.Google Scholar
Lin, B. L. S., ‘Elementary proofs of parity results for broken 3-diamond partitions’, J. Number Theory 135 (2014), 1–7.Google Scholar
Mahadeva Naika, M. S., Hemanthkumar, B. and Sumanth Bharadwaj, H. S., ‘Color partition identities arising from Ramanujan’s theta functions’, Acta Math. Vietnam. to appear.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.CrossRefGoogle Scholar
Ramanujan, S., Collected Papers (Cambridge University Press, Cambridge, 1927).Google Scholar
Watson, G. N., ‘Ramanujan’s Vermutung über Zerfällungsanzahlem’, J. reine angew. Math. 179 (1938), 97–128.Google Scholar
Xia, E. X. W. and Yao, O. X. M., ‘Parity results for 9-regular partitions’, Ramanujan J. 34 (2014), 109–117.CrossRefGoogle Scholar
Yao, O. X. M., ‘New parity results for broken 11-diamond partitions’, J. Number Theory 140 (2014), 267–276.Google Scholar