Some consequences of Glaisher's map and a generalization of Sylvester's theorem

TL;DR

This paper explores the properties of special classes of partitions, generalizes Sylvester's theorem, and provides new combinatorial interpretations of Rogers-Ramanujan identities using Glaisher's map.

Contribution

It introduces a new class of partitions generalizing self-conjugate partitions and extends Sylvester's theorem, offering novel combinatorial insights and interpretations.

Findings

Derived arithmetic properties of k-regular partitions with bounded parts

Extended Sylvester's theorem to a new class of partitions

Provided new combinatorial interpretations of Rogers-Ramanujan identities

Abstract

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.