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arXiv · 2410.16985

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

Shishuo Fu·Haijun Li
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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.

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Shishuo Fu, Haijun Li. 2024-10-22. Sequences of odd length in strict partitions II: the $2$-measure and refinements of Euler's theorem. https://arxiv.org/abs/2410.16985

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