A Partition Identity Related to Stanley's Theorem

Abstract

In this paper, we use the Lambert series generating function for Euler's totient function to introduce a new identity for the number of $1$'s in the partitions of $n$. A new expansion for Euler's partition function $p(n)$ is derived in this context. These surprising new results connect the famous classical totient function from multiplicative number theory to the additive theory of partitions.