发表机构
Univ Gustave Eiffel; Research Center for Mathematics and Interdisciplinary Sciences, Shandong University（巴黎-夏尔·戴高乐大学; 山东大学数学与交叉科学研究院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
通过在有界格路径上构造对合，建立了组合运行与广义返回统计量间的对称性，解决了有理Dyck路径的开放问题，并给出了$(a,b)$-停车函数的新生成函数。
AI 中文摘要
部分受停车函数及其变体枚举的启发，对由若干统计量（包括著名的 $\mathsf{run}$ 和 $\mathsf{return}$）细化的格路径的研究长期以来备受关注。我们考虑有界格路径，即由给定格路径界定且与向量停车函数相关的路径，并为此将 $\mathsf{run}$ 推广为组合运行（composition runs），由组合参数化。通过在这些路径上构造对合，我们建立了将组合运行与若干广义返回统计量联系起来的对称性。作为应用，我们解决了戴（Dai）、傅（Fu）和邱（Qiu）关于有理Dyck路径的一个开放问题。广义 $\mathsf{run}$ 与 $\mathsf{return}$ 之间的对称性随后被转移到向量停车函数上，这暗示了一个新的素数分解概念。在 $(a,b)$-停车函数的特殊情形下，我们计算了由 $\mathsf{run}$ 和 $\mathsf{pri}$ 细化的生成函数，该结果即使对经典停车函数也是新的。
英文摘要
Partly motivated by enumeration of parking functions and their variants, there is a long-standing interest in lattice paths refined by several statistics, including the notable $\mathsf{run}$ and $\mathsf{return}$. We consider bounded lattice paths, which are bounded by a given lattice path and are related to vector parking functions, for which we generalize $\mathsf{run}$ to composition runs, parameterized by a composition. By constructing involutions on these paths, we establish symmetries relating composition runs to some generalized return statistics. As an application, we settle an open problem of Dai, Fu, and Qiu on rational Dyck paths. The symmetries between generalized $\mathsf{run}$ and $\mathsf{return}$ are then transferred to vector parking functions, which suggest a new notion of prime decomposition. In the special case of $(a, b)$-parking functions, we compute the generating function refined by $\mathsf{run}$ and $\mathsf{pri}$, which is new even for classical parking functions.
Comments35 page, 16 figures
