## Abstract

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} ^{k}Δ^{m−1}x^{n−1}|_{x=rn−1}, where Δ is the difference operator and Δ_{r} is the difference operator of step r, that is, Δ_{r}p(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.

Original language | English |
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Pages (from-to) | 55-64 |

Number of pages | 10 |

Journal | Advances in Applied Mathematics |

Volume | 109 |

DOIs | |

State | Published - Aug 2019 |

### Bibliographical note

Publisher Copyright:© 2019 Elsevier Inc.

## ASJC Scopus subject areas

- Applied Mathematics