Bijections on $r$-Shi and $r$-Catalan Arrangements

Abstract

Associated with the $r$-Shi arrangement and $r$-Catalan arrangement in ${\mathbb{R}}^n$, 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.

arXiv comment19 pages, 2 figures

References in corpus (7)

• Product of Ginibre matrices: Fuss-Catalan and Raney distributions
• On generating functions of Hausdorff moment sequences
• Bell Polynomials and $k$-generalized Dyck Paths
• Counting unlabeled k-trees
• A New Bijection Between Forests and Parking Functions
• Counting generalized Dyck paths
• Note on the bijectivity of the Pak-Stanley labelling