Labeling the Regions of the Type $C_n$ Shi Arrangement

The number of regions of the type $C_n$ Shi arrangement in $\mathbb{R}^n$ is $(2n+1)^n$. Strikingly, no bijective proof of this fact has been given thus far. The aim of this paper is to provide such a bijection and use it to prove more refined results. We construct a bijection between the regions of the type $C_n$ Shi arrangement in $\mathbb{R}^n$ and sequences $a_1a_2 \ldots a_n$, where $a_i \in \{-n, -n+1, \ldots, -1, 0, 1, \ldots, n-1, n\}$, $ i \in [n]$. Our bijection naturally restrict to bijections between special regions of the arrangement and sequences with a given number of distinct elements.