2020/05/18 by Houshan Fu, Suijie Wang, Fu, Houshan +3
Engineering · Mathematics · #05A19 #52C35 #Advanced Combinatorial Mathematics #Advanced Mathematical Identities #Combinatorics (math.CO) #FOS: Mathematics #graph theory and CDMA systems
paper · pdf · doi:10.48550/arxiv.2005.08574
openalex publication_date 2020/05/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Associated with the r-Shi arrangement and r-Catalan arrangement in ℝn, 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.