## 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

Funding Information:The author thanks the referee for bringing the dissertation of Athanasiadis to his attention. The author also thanks Margaret Readdy for inspiring discussions and Richard Stanley and Thomas Zaslavsky for bringing the extended Shi arrangement to the author's attention. The author thanks Jennifer Rice for comments on an earlier version of this article. This research was partially supported by Swedish Natural Science Research Council DNR 702-238/98 . The author was also supported by National Science Foundation grants DMS 98-00910 and DMS 99-83660 while visiting Cornell University. This work was partially supported by a grant from the Simons Foundation (# 429370 to Richard Ehrenborg).

Publisher Copyright:

© 2019 Elsevier Inc.

## ASJC Scopus subject areas

- Applied Mathematics