Labeling the regions of the type \(C_n\) Shi arrangement (Q1953517)

Summary: 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.