按部分奇偶性限制的超着色分拆k元组
Overcolored Partition $k$-tuples Restricted by Parity of the Parts
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中文总结 AI 辅助
本文研究超着色分拆k元组的计数函数$\bar{b}^k_{r,s}(n)$，扩展Chacon与Sellers的相关结果，建立其模素数的可除性性质及模2幂的新同余式，所用技术基于theta函数恒等式和模形式。
中文摘要 AI 辅助
本文研究组合对象$\bar{b}^k_{r,s}(n)$，其计数的是超着色分拆k元组，其中偶数部分和奇数部分分别用r种和s种颜色着色。我们扩展了Chacon和Sellers针对多组r、s和k的结果，还建立了$\bar{b}^k_{r,s}(n)$模素数p的可除性性质，并以模2的幂的新同余式作为结论。获得结果所用的技术依赖于theta函数恒等式和模形式。
英文摘要
In this paper, we study the combinatorial object $\bar{b}^k_{r,s}(n)$ which counts the overcolored partition $k$-tuples wherein both even and odd parts are colored with $r$ and $s$ colors, respectively. We extend results of Chacon and Sellers for several families of $r,s$ and $k$. We also establish divisibility properties for $\bar{b}^k_{r,s}(n)$ modulo prime $p$ and conclude with new congruences modulo powers of $2$. The techniques involved to obtain our results rely on theta function identities and modular forms.
