Applications of Faà di Bruno's formula to partition traces

We revisit several partition-theoretic generating functions, including the theta quotients from Ramanujan’s lost notebook, MacMahon’s partition functions, and reciprocal sums of parts in partitions, through the lens of the classical Faà di Bruno formula. This approach offers a unified and natural reinterpretation of known results and provides a systematic framework for deriving new identities of a similar type.