AI 中文总结
本文证实了安德鲁斯和埃尔·巴赫拉维关于\(q\)级数\(T_o(q)\)系数奇偶性的猜想，还建立了\(S_1(q)\)系数模8的无穷多个同余式，并证明\(s_1(n)/8\)对\(n\geq0\)以自然密度1取整数值。
AI 中文摘要
最近，安德鲁斯和埃尔·巴赫拉维研究了一个分拆函数\(s_1(n)\)，它计算\(n\)的两色分拆中，最小部分仅出现在一种规定颜色中，而每个更大的部分可以出现在一种颜色、两种颜色或两种颜色都出现的不同部分的数量。他们得到了\(s_1(n)\)模4的完整描述。他们还考虑了一个\(q\)级数\(T_{o}(q)\)，它是\(s_1(n)\)生成函数的奇数伴随级数。在论文结尾，他们提出了关于\(T_o(q)\)系数奇偶性的猜想。本文证实了这个猜想。此外，我们建立了\(S_1(q)\)系数模8的无穷多个同余式，并证明对于\(n\geq0\)，\(s_1(n)/8\)以自然密度1取整数值。
英文摘要
Recently, Andrews and El Bachraoui studied a partition function $s_1(n)$, which counts the number of two-color partitions into distinct parts of $n$ whose smallest part occurs in one prescribed color only, while every larger part may occur in either color or in both colors. They obtained a complete description modulo 4 for $s_1(n)$. They also considered a $q$-series $T_{o}(q)$ which is the odd companion series of the generating function for $s_1(n)$. At the end of their paper, they presented a conjecture on the parity of the coefficients of $T_o(q)$. In this paper, we confirm this conjecture. Moreover, we establish an infinite family of congruences modulo 8 for the coefficients of $S_1(q)$ and prove that the set of integers satisfying $s_1(n)\equiv 0\pmod 8$ has natural density one.
