On the parity of the number of partitions with odd multiplicities

James A. Sellers, Fabrizio Zanello

Research output: Contribution to journalArticlepeer-review

Abstract

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)
JournalInternational Journal of Number Theory
DOIs
StateAccepted/In press - 2021

Bibliographical note

Publisher Copyright:
© 2021 World Scientific Publishing Company.

Copyright:
Copyright 2021 Elsevier B.V., All rights reserved.

Keywords

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

Fingerprint Dive into the research topics of 'On the parity of the number of partitions with odd multiplicities'. Together they form a unique fingerprint.

Cite this