## Abstract

Hetyei recently introduced a hyperplane arrangement (called the homogenized Linial arrangement) and used the finite field method of Athanasiadis to show that its number of regions is a median Genocchi number. These numbers count a class of permutations known as Dumont derangements. Here, we take a different approach, which makes direct use of Zaslavsky's formula relating the intersection lattice of this arrangement to the number of regions. We refine Hetyei's result by obtaining a combinatorial interpretation of the Möbius function of this lattice in terms of variants of the Dumont permutations. This enables us to derive a formula for the generating function of the characteristic polynomial of the arrangement. The Möbius invariant of the lattice turns out to be a (nonmedian) Genocchi number. Our techniques also yield type B, and more generally Dowling arrangement, analogs of these results.

Original language | English (US) |
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State | Published - 2019 |

Event | 31st International Conference on Formal Power Series and Algebraic Combinatorics, FPSAC 2019 - Ljubljana, Slovenia Duration: Jul 1 2019 → Jul 5 2019 |

### Conference

Conference | 31st International Conference on Formal Power Series and Algebraic Combinatorics, FPSAC 2019 |
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Country | Slovenia |

City | Ljubljana |

Period | 7/1/19 → 7/5/19 |

## Keywords

- Dowling lattices
- Genocchi numbers
- Hyperplane arrangements

## ASJC Scopus subject areas

- Algebra and Number Theory