Counting faces in the extended Shi arrangement

We define the level of a subset X of Euclidean space to be the dimension of the smallest subspace such that the distance between each element of X and the subspace is bounded. We prove that the number of faces in the n-dimensional extended Shi arrangement Aˆ_n ^r having codimension k and level m is given by m⋅(nk)⋅Δ_r ^kΔ^m−1xⁿ⁻¹|_x=rn−1, where Δ is the difference operator and Δ_r is the difference operator of step r, that is, Δ_rp(x)=p(x)−p(x−r). This generalizes a result of Athanasiadis which counts the number of faces of different dimensions in the extended Shi arrangement Aˆ_n ^r. The proof relies on a multi-variated Abel identity due to Françon.