Bijections on r-Shi and r-Catalan arrangements

This paper aims to establish two bijections: from regions of the r-Shi arrangement to O-rooted labeled r-trees, and from regions of the r-Catalan arrangement to pairings of permutation and r-Dyck path. To this end, we introduce a cubic matrix for each region of the hyperplane arrangements. The first bijection is established by reading the positions of minimal positive entries in its column slices. The second one is obtained by reading the numbers of positive entries in its column slice, which turns out to be essentially the same as the bijection obtained by Duarte and Guedes de Oliveira [10] in 2019. Moreover, by reading the numbers of positive entries in its row slices we will recover the Pak-Stanley labeling, which is a celebrated bijection from regions of the r-Shi arrangement to r-parking functions.