n色分拆的富兰克林恒等式及配套的贝克型恒等式
Franklin's identity for $n$-color partitions and companion Beck-type identities
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中文总结 AI 辅助
该研究针对n色分拆，证明了其对应的富兰克林定理与两个贝克型恒等式，并为所有定理提供了解析和组合两种证明方法。
中文摘要 AI 辅助
我们证明了普通分拆的一些经典恒等式在n色分拆中有精确类比，n色分拆指大小为n≥1的部分可出现颜色1,2,…,n的分拆。对r≥2且j≥0，记𝒪_{j,r}(m)为m的n色分拆中恰好有j个大小和颜色均被r整除的不同部分的集合，𝒟_{j,r}(m)为m的n色分拆中恰好有j个至少出现r次的不同部分的集合。我们证明了n色版本的富兰克林定理|𝒪_{j,r}(m)|=|𝒟_{j,r}(m)|，以及两个贝克型恒等式，对所有定理均给出了解析和组合两种证明。
英文摘要
We show that some classical identities valid for ordinary partitions have precise analogues for $n$-color partitions, that is partitions in which a part of size $n\geq 1$ can occur in colors $1, 2, \ldots, n$. For $r \ge 2$ and $j \ge 0$, we write $\mathcal{O}_{j,r}(m)$ and $\mathcal{D}_{j,r}(m)$ for the sets of $n$-color partitions of $m$ with, respectively, exactly $j$ different parts whose size and color are divisible by $r$, and exactly $j$ different parts occurring at least $r$ times. We prove an $n$-color version of Franklin's theorem, $|\mathcal{O}_{j,r}(m)| = |\mathcal{D}_{j,r}(m)|$, along with two Beck-type identities. We give both analytic and combinatorial proofs for all theorems.
