Bijections on r-Shi and r-Catalan arrangements

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