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.

中文翻译：

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关于奇数分区数的奇偶性

最近，Hirschhorn 和第一作者考虑了函数的奇偶性一种(n)它计算整数分区的数量n其中每个部分以奇数出现。他们得出了对平价的有效表征一种(2米)仅基于属性米.在本文中，我们快速验证了他们的结果，然后将其扩展到对奇偶校验的显式表征一种(n)对所有人n≢7(模组 8).我们还展示了从这些特征得出的一些无限的模 2 同余族。我们通过讨论案例来结束n ≡ 7(模组 8), 其中, 有趣的是, 的行为一种(n)模 2 似乎完全不同。特别是，我们推测，渐近，一种(8米 + 7)恰好是奇数50%的时间。这个猜想，其广泛推广到 eta-商的背景将成为后续论文的主题，仍然是广泛开放的。