Non-Rascoe partitions and a rank parity function associated to the Rogers-Ramanujan partitions

TL;DR

This paper explores the generating function related to Rogers-Ramanujan partitions, revealing connections to restricted partitions and deriving arithmetic properties, including conjectures on mod 4 congruences, with generalizations in a parameter.

Contribution

It introduces a new rank parity function for Rogers-Ramanujan partitions and links it to a novel class of restricted partitions, expanding understanding of their combinatorial and arithmetic properties.

Findings

Generated a new rank parity function for Rogers-Ramanujan partitions.

Connected the generating function to partitions into distinct parts with a non-part size.

Conjectured a mod 4 congruence for the number of such partitions.

Abstract

We study the generating function of the excess number of Rogers-Ramanujan partitions with odd rank over those with even rank, and, using combinatorial and analytical techniques, show that this generating function is closely connected with an interesting class of restricted partitions, namely, partitions into distinct parts where the number of parts is not a part. We derive arithmetic properties of the number of such partitions and conjecture an interesting mod $4$ congruence. Generalizations of most of these results in a parameter $\ell$ are also obtained.