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DIVISIBILITY OF CERTAIN SINGULAR OVERPARTITIONS BY POWERS OF $\textbf{2}$ AND $\textbf{3}$

Part of: Combinatorics Additive number theory; partitions

Published online by Cambridge University Press:  14 January 2021

AJIT SINGH Open the ORCID record for AJIT SINGH [Opens in a new window]  and
RUPAM BARMAN Open the ORCID record for RUPAM BARMAN [Opens in a new window]
AJIT SINGH
Affiliation:
Department of Mathematics, Indian Institute of Technology Guwahati, Assam, India, PIN-781039 e-mail: ajit18@iitg.ac.in
RUPAM BARMAN*
Affiliation:
Department of Mathematics, Indian Institute of Technology Guwahati, Assam, India, PIN-781039
*
e-mail: rupam@iitg.ac.in
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Abstract

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Andrews introduced the partition function $\overline {C}_{k, i}(n)$ , called the singular overpartition function, which counts the number of overpartitions of n in which no part is divisible by k and only parts $\equiv \pm i\pmod {k}$ may be overlined. We prove that $\overline {C}_{6, 2}(n)$ is almost always divisible by $2^k$ for any positive integer k. We also prove that $\overline {C}_{6, 2}(n)$ and $\overline {C}_{12, 4}(n)$ are almost always divisible by $3^k$ . Using a result of Ono and Taguchi on nilpotency of Hecke operators, we find infinite families of congruences modulo arbitrary powers of $2$ satisfied by $\overline {C}_{6, 2}(n)$ .

Keywords

arithmetic densityeta-quotientsmodular formssingular overpartitions

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 104 , Issue 2 , October 2021 , pp. 238 - 248
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Copyright
© 2021 Australian Mathematical Publishing Association Inc.

References

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