A refined view of a curious identity for partitions into odd parts with designated summands

TL;DR

This paper explores a refined perspective on a special class of integer partitions called partitions with designated summands, focusing on those with only odd parts, and introduces new combinatorial insights into their structure.

Contribution

The paper provides a new refined view and combinatorial interpretation of partitions with designated summands, especially for odd parts, expanding understanding of their properties.

Findings

New combinatorial identities for partitions with designated summands.

Enhanced understanding of partitions restricted to odd parts.

Potential applications to partition theory and q-series.

Abstract

In 2002, Andrews, Lewis, and Lovejoy introduced the combinatorial objects which they called partitions with designated summands. These are constructed by taking unrestricted integer partitions and designating exactly one of each occurrence of a part. In the same work, they also considered the restricted partitions with designated summands wherein all parts must be odd, and they denoted the corresponding function by $\mathrm{PDO}(n)$.