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 note, we quickly reprove their result, and then extend it to an explicit characterization of the parity of a(n)
Nearby in the stack
for all
n≡7(mod8).
We also exhibit some infinite families of congruences modulo 2 which follow from these characterizations. We conclude by discussing the case
n≡7(mod8)
, 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.
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