Applications of Fa\`{a} di Bruno's formula to partition traces

TL;DR

This paper uses Faà di Bruno's formula to unify and extend the understanding of various partition-theoretic generating functions, including those from Ramanujan and MacMahon, offering new identities and insights.

Contribution

It introduces a systematic framework applying Faà di Bruno's formula to partition functions, unifying known results and enabling derivation of new identities.

Findings

Unified reinterpretation of partition generating functions

Systematic derivation of new identities

Enhanced understanding of theta quotients and reciprocal sums

Abstract

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 Fa\`{a} 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.