Bounds on the Möbius-signed partition numbers

For $n\in \mathbb{N}$ let $\mathrm{\Pi }[n]$ denote the set of partitions of $n$, i.e., the set of positive integer tuples $(x_1,x_2,\dots ,x_k)$ such that $x_1\ge x_2\ge \cdots \ge x_k$ and $x_1+x_2+\cdots +x_k=n$. Fixing $f:\mathbb{N}\to \{0,\pm 1\}$, for $\pi =(x_1,x_2,\dots ,x_k)\in \mathrm{\Pi }[n]$ let $f(\pi ):=f(x_1)f(x_2)\cdots f(x_k)$. In this way we define the {signed partition numbers} $$p(n,f)=\sum _{\pi \in \mathrm{\Pi }[n]}f(\pi )\mathrm{.}$$ Following work of Vaughan and Gafni on partitions into primes and prime powers, we derive asymptotic formulae for quantities $p(n,\mu )$ and $p(n,\lambda )$, where $\mu$ and $\lambda$ denote the Möbius and Liouville functions from prime number theory, respectively. In addition we discuss how quantities $p(n,f)$ generalize the classical notion of restricted partitions.