On the area-depth symmetry on Łukasiewicz paths

In an effort to further understanding $q, t$ -Catalan statistics, a new statistic on Dyck paths called $d e p t h$ was proposed in Pappe, Paul and Schilling (2022) and was shown to be jointly equi-distributed with the well-known $a r e a$ statistics. In a recent preprint, Qu and Zhang (2025) generalized $d e p t h$ to so-called `` $\vec{k}$ -Dyck paths’'. They showed that $a r e a$ and $d e p t h$ 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 $a r e a$ and $d e p t h$ under the classical bijection between Łukasiewicz paths and plane trees, through which the symmetry is transparent.

On the area-depth symmetry on Łukasiewicz paths

文献摘要

讨论

讲解博客

AI智能校验

On the area-depth symmetry on Łukasiewicz paths

文献摘要

讨论

讲解博客

AI智能校验

On the area-depth symmetry on Łukasiewicz paths

文献摘要

讨论

|讲解博客

|AI智能校验

On the area-depth symmetry on Łukasiewicz paths

文献摘要

讨论

|讲解博客

|AI智能校验

讲解博客

AI智能校验

讲解博客

AI智能校验