Counting faces in the extended Shi arrangement ${\widehat{\mathcal A}}_{n}^{r}$

Richard Ehrenborg

To appear at Formal Power Series and Algebraic Combinatorics (FPSAC01), Tempe, Arizona (USA), May 20-26, 2001


Abstract

We define the {\em 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 ${\widehat{\mathcal A}}_{n}^{r}$ having codimension $k$ and level $m$ is given by $m \cdot {n \choose k} \cdot \Delta_{r}^{k} \Delta^{m-1} x^{n-1} \left| {_{x = r n - 1}} \right.$ where $\Delta$ is the difference operator and $\Delta_{r}$ is the difference operator of step $r$, that is, $\Delta_{r} p(x) = p(x) - p(x-r)$. This generalizes a result of Athanasiadis which counts the number of faces of different dimensions from the Shi arrangement~${\widehat{\mathcal A}}_{n}$. The proof relies on extending Athanasiadis' result to ${\widehat{\mathcal A}}_{n}^{r}$ and applying a multi-variated Abel identity.


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