A simple bijection for the regions of the Shi arrangement of hyperplanes

Abstract The Shi arrangement Ln is the arrangement of affine hyperplanes in R n of the form xi − xj = 0 or 1, for 1 ⩽ i R n into (n + 1)n−1 regions, as was first proved by Shi. We give a simple bijective proof of this result. Our bijection generalizes easily to any subarrangement of Ln containing the hyperplanes xi − xj = 0 and to the extended Shi arrangements. It also implies the fact that the number of regions of Ln which are relatively bounded is (n − 1)n−1.

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