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| Format: | Preprint |
| Published: |
2026
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2603.23879 |
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Table of Contents:
- In a remarkable paper, Tatsuyuki Hikita settled a longstanding e-positivity conjecture of Stanley and Stembridge. Among many other things, he wrote down a certain formula $φ_k$, and proved that the $φ_k$ sum to one, thereby defining a probability distribution. Though Hikita's proof was simple, it remains surprising that the $φ_k$ sum to one. In this note, we give a combinatorial interpretation of Hikita's probability distribution. The main tool is a certain permutation statistic that we call the watershed. After seeing an early version of our work, Darij Grinberg noticed that the permutation statistic was implicit in a so-called "bulldozer problem" that was on the short list for the 2015 International Mathematics Olympiad. However, our description of the statistic, which makes use of the Renyi-Foata bijection, appears to be new.