互质分拆与Jordan函数
Coprime partitions and Jordan totient functions
摘要：
我们表明，虽然互质的组合物的正整数$n$到$k$部分的数量可以表示为一个$\mathbb{Q}$ -线性组合的约旦totient功能，这是永远不可能的互质分区的$n$到$k$部分。我们还表明，$n$到$k$部分的互质分区数$p_k'(n)$可以表示为$\mathbb{C}$ -Jordan totient函数的线性组合，对于$n$足够大，当且仅当$k\in \{2,3\}$和以一种独特的方式。最后，我们介绍了Jordan函数的一些推广，并证明了$p_k'(n)$总是可以表示为它们的$\mathbb{C}$ -线性组合。
英文摘要：
We show that while the number of coprime compositions of a positive integer $n$ into $k$ parts can be expressed as a $\mathbb{Q}$-linear combinations of the Jordan totient functions, this is never possible for the coprime partitions of $n$ into $k$ parts. We also show that the number $p_k'(n)$ of coprime partitions of $n$ into $k$ parts can be expressed as a $\mathbb{C}$-linear combinations of the Jordan totient functions, for $n$ sufficiently large, if and only if $k\in \{2,3\}$ and in a unique way. Finally we introduce some generalizations of the Jordan totient functions and we show that $p_k'(n)$ can be always expressed as a $\mathbb{C}$-linear combinations of them.
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