On the Homogenized Linial Arrangement: Intersection Lattice and Genocchi Numbers

Alexander Lazar and Michelle L. Wachs

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. 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.

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