Partition identities associated to Rogers–Ramanujan type identities

An analytical Rogers–Ramanujan type identity is an identity between an infinite product and an infinite series that are generating functions of suitable sets of partitions. We show that, in many cases, there are infinitely many sets of partitions corresponding to a single analytical Rogers–Ramanujan type identity. We give an explicit description of these infinite sets explaining how to interpret the sum side combinatorially via generating functions. We also describe the bijection between these sets of partitions. Moreover, we derive a new infinite family of analytical Rogers–Ramanujan type identities associated with Glaisher’s classical partition theorems, together with infinitely many partition-theoretic interpretations linked by explicit bijections.