Bounds on the M\"obius-signed partition numbers

For $n \in \mathbb{N}$ let $\Pi[n]$ denote the set of partitions of $n$, i.e., the set of positive integer tuples $(x_1,x_2,\ldots,x_k)$ such that $x_1 \geq x_2 \geq \cdots \geq x_k$ and $x_1 + x_2 + \cdots + x_k = n$. Fixing $f:\mathbb{N}\to\{0,\pm 1\}$, for $\pi = (x_1,x_2,\ldots,x_k) \in \Pi[n]$ let $f(\pi) := f(x_1)f(x_2)\cdots f(x_k)$. In this way we define the {signed partition numbers} $$p(n,f) = \sum_{\pi\in\Pi[n]} f(\pi).$$ 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 M\"obius and Liouville functions from prime number theory, respectively. In addition we discuss how quantities $p(n,f)$ generalize the classical notion of restricted partitions.