论文标题
$ r $ -shi和$ r $ -CATALAN安排的双线
Bijections on $r$-Shi and $r$-Catalan Arrangements
论文作者
论文摘要
与$ r $ -SHI布置和$ r $ -CATALAN在$ \ bbb {r}^n $中的布置相关联，我们为每个区域引入一个立方矩阵，以统一的方式建立了两个两种素质。首先，在立方矩阵的列切片中，最小正项的位置将对$ r $ -SHI安排的区域进行两次培训，以归因于$ o $ $ $ $ $ $ $ $ r $ $ -trees。其次，立方矩阵的列切片中的积极条目数量将对$ r $ -CATALAN安排的区域进行两次射击，以置于排列配对和$ r $ -DYCK路径。此外，立方矩阵的行片段中的积极条目数量将恢复Pak-Stanley标签，这是从$ R $ -SHI安排到$ r $ $ parkarking功能的著名的两者。
Associated with the $r$-Shi arrangement and $r$-Catalan arrangement in $\Bbb{R}^n$, we introduce a cubic matrix for each region to establish two bijections in a uniform way. Firstly, the positions of minimal positive entries in column slices of the cubic matrix will give a bijection from regions of the $r$-Shi arrangement to $O$-rooted labeled $r$-trees. Secondly, the numbers of positive entries in column slices of the cubic matrix will give a bijection from regions of the $r$-Catalan arrangement to pairings of permutation and $r$-Dyck path. Moreover, the numbers of positive entries in row slices of the cubic matrix will recover the Pak-Stanley labeling, a celebrated bijection from regions of the $r$-Shi arrangement to $r$-parking functions.
