Block odd index and its variations

Authors

• Runqiao Li, The University of Texas Rio Grande Valley
• Andrew Y.Z. Wang, University of Electronic Science and Technology of China
• Zheng Xu, Pennsylvania State University

Document Type

Article

Publication Date

9-1-2026

Abstract

Sylvester’s fish-hook bijection maps the length of partitions into odd parts to the alternating sum of partitions into distinct parts. It is natural to ask which statistic of partitions into odd parts will correspond to the length of partitions into distinct parts. One can hardly believe that there actually exists such one. In this paper, we first define a simple statistic, called block odd index, to address this question. Then we extend the results to ordinary partitions through the weighted block odd index. Finally, we introduce the Durfee odd rank to connect the block odd index with the 2-modular diagram of partitions into odd parts.

Comments

https://rdcu.be/9VLYPPWCau8a

Recommended Citation

Li, Runqiao, Andrew YZ Wang, and Zheng Xu. "Block odd index and its variations." The Ramanujan Journal 71, no. 1 (2026): 3. https://doi.org/10.1007/s11139-026-01456-9

Publication Title

Ramanujan Journal

DOI

10.1007/s11139-026-01456-9