AI 中文总结
本文研究非对称Catalan与Shi超平面排列的区域，利用有限域方法得到其特征多项式与区域数公式，引入两类新组合对象并建立双射，解决了研讨会上的问题，推广了Stanley的对应关系。
AI 中文摘要
本文研究Catalan排列与Shi排列的非对称扩展的区域。我们首先利用有限域方法确定一类一般的Catalan排列与Shi排列的特征多项式，从而得到区域数的显式公式。随后，我们引入两类新的组合对象：c标记Dyck路径与c构建函数，二者分别推广经典Dyck路径与停车函数。我们证明非对称Catalan排列的区域与c标记Dyck路径存在自然双射，非对称Shi排列的区域与c构建函数存在自然双射。这解决了Theo Douvropoulos在2022年奥伯沃尔夫赫枚举组合学研讨会上提出的问题。特别地，我们的结果通过引入新的组合模型并采用完全不同的证明技巧，推广了Stanley提出的m停车函数与m Shi排列区域之间的著名对应关系。
英文摘要
In this paper, we study the regions of asymmetric extensions of the Catalan and Shi arrangements. We first determine the characteristic polynomials using the finite field method, thereby obtaining explicit formulas for the number of regions. We then introduce two new classes of combinatorial objects: $\mathbf{c}$-labelled Dyck paths and $\mathbf{c}$-building functions, which generalize classical Dyck paths and parking functions, respectively. We prove that the regions of the asymmetric $\mathbf{c}$-Catalan arrangement are in natural bijection with $\mathbf{c}$-labelled Dyck paths, while those of the asymmetric $\mathbf{c}$-Shi arrangement are in natural bijection with $\mathbf{c}$-building functions. This resolves a problem proposed by Theo Douvropoulos during the 2022 Oberwolfach Workshop on Enumerative Combinatorics. In particular, our results extend Stanley's celebrated correspondence between $k$-parking functions and the regions of the $k$-Shi arrangement by introducing new combinatorial models and developing entirely different proof techniques.
Comments36 pages, this is Version 2, which includes the more general case. We have also removed Question 3 from the open issues. Suggestions and feedback are welcome
