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POWER PARTITIONS AND SEMI-$m$-FIBONACCI PARTITIONS

Part of: Combinatorics Additive number theory; partitions

Published online by Cambridge University Press:  20 February 2020

ABDULAZIZ M. ALANAZI Open the ORCID record for ABDULAZIZ M. ALANAZI [Opens in a new window] ,
AUGUSTINE O. MUNAGI Open the ORCID record for AUGUSTINE O. MUNAGI [Opens in a new window]  and
DARLISON NYIRENDA Open the ORCID record for DARLISON NYIRENDA [Opens in a new window]
ABDULAZIZ M. ALANAZI
Affiliation:
Department of Mathematics, Faculty of Sciences, University of Tabuk, P.O. Box 741, Tabuk 71491, Saudi Arabia email am.alenezi@ut.edu.sa
AUGUSTINE O. MUNAGI
Affiliation:
School of Mathematics, University of the Witwatersrand, P.O. Wits 2050, Johannesburg, South Africa email augustine.munagi@wits.ac.za
DARLISON NYIRENDA*
Affiliation:
School of Mathematics, University of the Witwatersrand, P.O. Wits 2050, Johannesburg, South Africa email darlison.nyirenda@wits.ac.za
*
✉darlison.nyirenda@wits.ac.za
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Abstract

Andrews [‘Binary and semi-Fibonacci partitions’, J. Ramanujan Soc. Math. Math. Sci.7(1) (2019), 1–6] recently proved a new identity between the cardinalities of the set of semi-Fibonacci partitions and the set of partitions into powers of 2 with all parts appearing an odd number of times. We extend the identity to the set of semi-$m$-Fibonacci partitions of $n$ and the set of partitions of $n$ into powers of $m$ in which all parts appear with multiplicity not divisible by $m$. We also give a new characterisation of semi-$m$-Fibonacci partitions and some congruences satisfied by the associated number sequence.

Keywords

partitionbijectioncongruence

MSC classification

Primary: 11P84: Partition identities; identities of Rogers-Ramanujan type
Secondary: 11P83: Partitions; congruences and congruential restrictions 05A15: Exact enumeration problems, generating functions
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 102 , Issue 3 , December 2020 , pp. 418 - 429
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Copyright
© 2020 Australian Mathematical Publishing Association Inc.

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References

Andrews, G. E., ‘Binary and semi-Fibonacci partitions’, volume honouring A. K. Agarwal’s 70th birthday, J. Ramanujan Soc. Math. Math. Sci. 7(1) (2019), 1–6.Google Scholar
Andrews, G. E., The Theory of Partitions (Addison-Wesley, Reading, MA, 1976), reprinted, Cambridge University Press, Cambridge, 1984, 1998.Google Scholar
Beck, G., ‘Semi-Fibonacci partitions’, Wolfram Demonstrations Project, 2015,http://demonstrations.wolfram.com/SemiFibonacciPartitions/.Google Scholar
Gafni, A., ‘Power partitions’, J. Number Theory 163 (2016), 19–42.CrossRefGoogle Scholar
Sloane, N. J. A., ‘The on-line encyclopedia of integer sequences’, 2015, available on-line at http://oeis.org.Google Scholar