On the parity of the number of partitions with odd multiplicities
Document Type
Article
Publication Date
2-26-2021
Department
Department of Mathematical Sciences
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.
Publication Title
International Journal of Number Theory
Recommended Citation
Sellers, J.,
&
Zanello, F.
(2021).
On the parity of the number of partitions with odd multiplicities.
International Journal of Number Theory.
http://doi.org/10.1142/S1793042121500573
Retrieved from: https://digitalcommons.mtu.edu/michigantech-p/14744