On the Homogenized Linial Arrangement: Intersection Lattice and Genocchi Numbers

TL;DR

This paper explores the intersection lattice of the homogenized Linial arrangement, linking it to Genocchi numbers and Dumont permutations, and extends results to type B and Dowling arrangements.

Contribution

It provides a combinatorial interpretation of the M"obius function of the intersection lattice and derives formulas for the characteristic polynomial, extending to type B and Dowling arrangements.

Findings

The M"obius invariant of the lattice is a nonmedian Genocchi number.

A formula for the generating function of the characteristic polynomial is obtained.

Extensions to type B and Dowling arrangements are established.

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 M\"obius 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 characterisitic polynomial of the arrangement. The M\"obius 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.