On the area-depth symmetry on Łukasiewicz paths
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arXiv
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866918529339490304 |
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| author | Fang, Wenjie |
| author_facet | Fang, Wenjie |
| contents | In an effort to further understanding $q,t$-Catalan statistics, a new statistic on Dyck paths called $\mathtt{depth}$ was proposed in Pappe, Paul and Schilling (2022) and was shown to be jointly equi-distributed with the well-known $\mathtt{area}$ statistics. In a recent preprint, Qu and Zhang (2025) generalized $\mathtt{depth}$ to so-called ``$\vec{k}$-Dyck paths''. They showed that $\mathtt{area}$ and $\mathtt{depth}$ are also jointly equi-distributed over such paths with a fixed multiset of up-steps and a given first up-step, and they conjectured that the same holds when also fixing the last up-step. In this short note, we settle this conjecture on the more general context of Łukasiewicz paths by interpreting $\mathtt{area}$ and $\mathtt{depth}$ under the classical bijection between Łukasiewicz paths and plane trees, through which the symmetry is transparent. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_17949 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On the area-depth symmetry on Łukasiewicz paths Fang, Wenjie Combinatorics In an effort to further understanding $q,t$-Catalan statistics, a new statistic on Dyck paths called $\mathtt{depth}$ was proposed in Pappe, Paul and Schilling (2022) and was shown to be jointly equi-distributed with the well-known $\mathtt{area}$ statistics. In a recent preprint, Qu and Zhang (2025) generalized $\mathtt{depth}$ to so-called ``$\vec{k}$-Dyck paths''. They showed that $\mathtt{area}$ and $\mathtt{depth}$ are also jointly equi-distributed over such paths with a fixed multiset of up-steps and a given first up-step, and they conjectured that the same holds when also fixing the last up-step. In this short note, we settle this conjecture on the more general context of Łukasiewicz paths by interpreting $\mathtt{area}$ and $\mathtt{depth}$ under the classical bijection between Łukasiewicz paths and plane trees, through which the symmetry is transparent. |
| title | On the area-depth symmetry on Łukasiewicz paths |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2601.17949 |