The partition function $p(n)$ in terms of the classical Möbius function

In this paper, we investigate decompositions of the partition function $p(n)$ from the additive theory of partitions considering the famous Möbius function $\mu (n)$ from multiplicative number theory. Some combinatorial interpretations are given in this context. Our work extends several analogous identities proved recently relating $p(n)$ and Euler’s totient function $\phi (n)$. Keywords: Lambert series; Möbius function; $q$-series; partition function