Sequences of odd length in strict partitions II: the $2$-measure and refinements of Euler's theorem

TL;DR

This paper explores the number of odd-length sequences in strict partitions, linking it to the recently introduced 2-measure, and presents new refinements of Euler's partition theorem using advanced combinatorial tools.

Contribution

It establishes a connection between the sequence count and the 2-measure, provides a new $q$-series identity, and introduces two novel refinements of Euler's theorem with new combinatorial concepts.

Findings

Derived a $q$-series identity via three methods, including a Franklin-type involution.

Revisited Euler's partition theorem with new bivariate refinements.

Introduced new combinatorial notions such as the alternating index of partitions.

Abstract

The number of sequences of odd length in strict partitions (denoted as $\mathrm{sol}$), which plays a pivotal role in the first paper of this series, is investigated in different contexts, both new and old. Namely, we first note a direct link between $\mathrm{sol}$ and the $2$-measure of strict partitions when the partition length is given. This notion of $2$-measure of a partition was introduced quite recently by Andrews, Bhattacharjee, and Dastidar. We establish a $q$-series identity in three ways, one of them features a Franklin-type involuion. Secondly, still with this new partition statistic $\mathrm{sol}$ in mind, we revisit Euler's partition theorem through the lens of Sylvester-Bessenrodt. Two new bivariate refinements of Euler's theorem are established, which involve notions such as MacMahon's 2-modular Ferrers diagram, the Durfee side of partitions, and certain alternating index of partitions that we believe is introduced here for the first time.