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arXiv · 2603.23879

Foata, Hikita, and the Bulldozer Problem

Timothy Y. Chow
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Abstract

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 ${\varphi}_k$, and proved that the ${\varphi}_k$ sum to one, thereby defining a probability distribution. Though Hikita's proof was simple, it remains surprising that the ${\varphi}_k$ 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.

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Timothy Y. Chow. 2026-03-25. Foata, Hikita, and the Bulldozer Problem. https://arxiv.org/abs/2603.23879

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