论文标题
关于具有奇数多重性的分区数量的奇偶校验
On the parity of the number of partitions with odd multiplicities
论文作者
论文摘要
最近，Hirschhorn和第一作者考虑了函数$ a（n）$的均衡，该功能$ a（n）$计算了$ n $的整数分区的数量，其中每个部分都以奇数多种形式出现。它们得出了仅基于本注释中$m。$的属性的均衡表征的有效表征，我们迅速谴责他们的结果，然后将其扩展到对所有$ n \ equiv 7 \ equiv 7 \ pmod {8}的均等表征的明确表征。 最后，我们讨论了$ n \ equiv 7 \ pmod {8} $的情况，有趣的是，$ a（n）$ modulo 2的行为似乎完全不同。特别是，我们猜想，零散的$ a（800万+7）$是奇怪的时间$ 50 \％$。这种猜想的广泛概括是随后的论文的主题，它仍然是敞开的。
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 $a(n)$ for all $n\not\equiv 7 \pmod{8}.$ We also exhibit some infinite families of congruences modulo 2 which follow from these characterizations. We conclude by discussing the case $n\equiv 7 \pmod{8}$, 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.
