Generalizations of Sylvester's refinement of Euler's odd-strict theorem

TL;DR

This paper explores the periodicity properties in Bessenrodt's generalization of Sylvester's refinement of Euler's odd-strict theorem, extending the understanding of partition identities beyond the original scope.

Contribution

It investigates the periodicity of exponents in Bessenrodt's generalization and introduces a new partition identity that disregards this periodicity.

Findings

Identified periodicity patterns in r-regular partitions

Generalized the partition identity beyond periodicity constraints

Enhanced understanding of Sylvester's refinement in partition theory

Abstract

Regarding Euler's odd-strict theorem, which is the most basic partition identity, A refinement was done by Sylvester, and it was generalized by Bessenrodt to the r-regular and r-class regular cases. In this paper, we focus on the periodicity of the exponent seen on the r-regular side in Bessenrodt's generalization, and further generalize from that point of view. At the end of this paper, we also introduce a partition identity when extended without regard to the periodicity of the exponent.