区域 $\defc\le\min(\area,\dinv)$ 中交换 area 和 dinv 的 Dyck 路径上的对合
An involution on Dyck paths in the region $\defc\le\min(\area,\dinv)$ that interchanges area and dinv
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中文总结 AI 辅助
本文构造了满足 $\defc\le\min(\area,\dinv)$ 的 Dyck 路径上的显式对合，交换 area 和 dinv，基于新组合对象及双 Dyck 插入和 Garsia--Milne 原理。
中文摘要 AI 辅助
我们构造了一个满足 $\defc\le\min(\area,\dinv)$ 的 Dyck 路径上的显式对合，该对合交换 area 和 dinv。该构造依赖于本文开发的若干新组合对象，以及两个主要外部工具：\cite{Hawkes26} 的双 Dyck 插入和 Doyle~\cite{Doyle19} 所表述的 Garsia--Milne 对合原理~\cite{GarsiaMilne81}。
英文摘要
We construct an explicit involution on Dyck paths satisfying $\defc\le\min(\area,\dinv)$ that interchanges area and dinv, where $\defc=\binom n2-\area-\dinv$ and $n$ is the semilength. In addition, we give an involution on Dyck paths with $\defc \le 2n-8$ that interchanges area and dinv. Finally, we give an explicit partition formula for the portion of the $q,t$-Catalan polynomial of total degree at least $\binom n2-2n+8$, thereby proving a conjecture of Lee and Li~\cite[Conjecture~4]{LeeLi11}. Altogether our results give a combinatorial explanation of $q,t$-Catalan symmetry in the region $\defc\le\max(2n-8,\min(\area,\dinv))$. Our constructions rely on a number of new combinatorial objects developed here and three main external tools: the dual Dyck insertion of \cite{Hawkes26}, the Garsia--Milne involution principle~\cite{GarsiaMilne81},~\cite{Doyle19}, and the partition bijection of ~\cite{LoehrWarrington09}.
