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

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.