Labeling the Regions of the Type Cn Shi Arrangement

The number of regions of the type Cn Shi arrangement in Rn 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 Cn Shi arrangement in Rn and sequences a1a2 . . . an, where ai ∈ {−n,−n+1, . . . ,−1, 0, 1, . . . , n−1, n}, i ∈ [n]. Our bijection naturally restrict to bijections between special regions of the arrangement and sequences with a given number of distinct elements.