Guardado en:
| Autor principal: | |
|---|---|
| Formato: | Preprint |
| Publicado: |
2026
|
| Materias: | |
| Acceso en línea: | https://arxiv.org/abs/2603.23879 |
| Etiquetas: |
Agregar Etiqueta
Sin Etiquetas, Sea el primero en etiquetar este registro!
|
| _version_ | 1866917397788622848 |
|---|---|
| author | Chow, Timothy Y. |
| author_facet | Chow, Timothy Y. |
| 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_23879 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Foata, Hikita, and the Bulldozer Problem Chow, Timothy Y. Combinatorics 05A05 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. |
| title | Foata, Hikita, and the Bulldozer Problem |
| topic | Combinatorics 05A05 |
| url | https://arxiv.org/abs/2603.23879 |