Bijections on -Shi and -Catalan Arrangements
arXiv:2005.08574
Abstract
Associated with the -Shi arrangement and -Catalan arrangement in , 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 -Shi arrangement to -rooted labeled -trees. Secondly, the numbers of positive entries in column slices of the cubic matrix will give a bijection from regions of the -Catalan arrangement to pairings of permutation and -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 -Shi arrangement to -parking functions.
19 pages, 2 figures