  For positive integers $k, l \geq 2$, the set of $k$-regular partitions in
which parts appear at most $l$ times has attracted a lot of interest in that a
composition of Glaisher's mapping can be used to prove the associated partition
identities in certain cases. We consider some special cases and derive some
arithmetic properties. Of particular focus is the set of partitions in which
parts are odd and distinct ($k =2$, $l = 2$). Sylvester proved that, for fixed
weight, this set of partitions is equinumerous with the set of self-conjugate
partitions. We introduce a new class of partitions that generalizes
self-conjugate partitions and as a result, we extend Sylvester's theorem.
Furthermore, using this class of partitions, we give new combinatorial
interpretation of some Rogers-Ramanujan identities which were previously
considered by A. K. Agarwal.
