On the parity of the number of partitions with odd multiplicities

James A. Sellers, Fabrizio Zanello

Research output: Contribution to journalArticlepeer-review

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Recently, Hirschhorn and the first author considered the parity of the function a(n) which counts the number of integer partitions of n wherein each part appears with odd multiplicity. They derived an effective characterization of the parity of a(2m) based solely on properties of m. In this paper, we quickly reprove their result, and then extend it to an explicit characterization of the parity of a(n) for all n =7(mod 8). We also exhibit some infinite families of congruences modulo 2 which follow from these characterizations. We conclude by discussing the case n 7(mod 8), where, interestingly, the behavior of a(n) modulo 2 appears to be entirely different. In particular, we conjecture that, asymptotically, a(8m + 7) is odd precisely 50% of the time. This conjecture, whose broad generalization to the context of eta-quotients will be the topic of a subsequent paper, remains wide open.

Original languageEnglish (US)
Pages (from-to)1717-1728
Number of pages12
JournalInternational Journal of Number Theory
Issue number7
StatePublished - Aug 2021

Bibliographical note

Funding Information:
The idea of this work originated during a visit of the first author to Michigan Tech in Spring 2020. The second author thanks his former PhD student Samuel Judge and William Keith for several discussions on the broader topic of this paper. The second author was partially supported by a Simons Foundation grant (#630401).

Publisher Copyright:
© 2021 World Scientific Publishing Company.


  • Partition function
  • binary integer representation
  • density odd values
  • eta-quotient
  • odd multiplicity


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