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arXiv 1004.0547math.CO
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奇部相异的双分拆的同余

Congruences for Bipartitions with Odd Parts Distinct

William Y. C. Chen, Bernard L. S. Lin

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15 pages
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摘要：

Hirschhorn和Sellers研究了奇数部分不同的分区数的算术性质。在另一个方向上，哈蒙德和刘易斯研究了二等分数的算术性质。本文考虑奇部相异的双分拆的个数。让这个数字表示为$pod_{-2}(n)$。我们得到了$pod_{-2}(n)$的两个Ramanujan型恒等式，这意味着$pod_{-2}(2n+1)$是偶数，而$pod_{-2}(3n+2)$可被3整除.此外，我们表明，对于任何$α\geq 1$和$n\geq 0$，$ pod_{-2}(3^{2α+1}n+\frac{23\times 3^{2α}-7}{8})$是3的倍数和$pod_{-2}(5^{α+1}n+\frac{11\times 5^α+1}{4})$是整除5。我们还发现组合解释的两个同余模2和3。

英文摘要：

Hirschhorn and Sellers studied arithmetic properties of the number of partitions with odd parts distinct. In another direction, Hammond and Lewis investigated arithmetic properties of the number of bipartitions. In this paper, we consider the number of bipartitions with odd parts distinct. Let this number be denoted by $pod_{-2}(n)$. We obtain two Ramanujan type identities for $pod_{-2}(n)$, which imply that $pod_{-2}(2n+1)$ is even and $pod_{-2}(3n+2)$ is divisible by 3. Furthermore, we show that for any $α\geq 1$ and $n\geq 0$, $ pod_{-2}(3^{2α+1}n+\frac{23\times 3^{2α}-7}{8})$ is a multiple of 3 and $pod_{-2}(5^{α+1}n+\frac{11\times 5^α+1}{4})$ is divisible by 5. We also find combinatorial interpretations for the two congruences modulo 2 and 3.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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