Hostname: page-component-cd9895bd7-p9bg8 Total loading time: 0 Render date: 2024-12-28T14:39:01.024Z Has data issue: false hasContentIssue false
  • English
  • Français

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
Get access Rights & Permissions [Opens in a new window]

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
Check for updates
Copyright
© 2020 Australian Mathematical Publishing Association Inc.

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

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