Partition identities associated to Rogers-Ramanujan type identities

TL;DR

This paper explores the combinatorial structures behind Rogers-Ramanujan type identities, revealing infinite partition sets linked to each identity and providing explicit descriptions and new identities via Glaisher's transformations.

Contribution

It demonstrates the existence of infinitely many partition sets associated with a single Rogers-Ramanujan type identity and offers explicit combinatorial interpretations and new identities.

Findings

Infinite sets of partitions correspond to each identity.

Explicit combinatorial descriptions of sum sides.

New identities derived from Glaisher's identities.

Abstract

We show that, in many cases, there are infinitely many sets of partitions corresponding to a single analytical Rogers-Ramanujan type identity. This means that a single analytical Rogers-Ramanujan type identity implies the existence of bijections among infinitely many sets of partitions. We also give an explicit description of these infinite sets coming from the sum side of the analytical identity explaining how to interpret the sum side combinatorially as the generating function of the partitions considered. Moreover, we give a new infinite familiy of Rogers-Ramanujan type identities obtained by the Glaisher's identities.