感觉不如传统派

前情提要：https://www.cnblogs.com/CDOI-24374/p/19014029

你说的对但是

\[\begin{aligned} L(n)&=\prod\limits_{S\subseteq[n]}\left(\gcd\limits_{i\in S}i\right)^{(-1)^{|S|+1}}\\ &=\prod\limits_{d=1}^nd^{\sum\limits_{S\subseteq[n/d]}\left[\gcd\limits_{i\in S}i=1\right](-1)^{|S|+1}}\\ &=\prod\limits_{d=1}^nd^{\sum\limits_{S\subseteq[n/d]}(-1)^{|S|+1}\sum\limits_{k\mid S_i}\mu(k)}\\ &=\prod\limits_{d=1}^nd^{\sum\limits_{k=1}^{n/d}\mu(k)\sum\limits_{S\subseteq[n/dk]}(-1)^{|S|+1}}\\ &=\prod\limits_{d=1}^nd^{\sum\limits_{k=1}^{n/d}\mu(k)}\\ &=\prod\limits_{d=1}^n\prod\limits_{k=1}^{n/d}d^{\mu(k)}\\ &=\prod\limits_{k=1}^n(n/k)!^{\mu(k)} \end{aligned} \]