Séminaire Lotharingien de Combinatoire, 82B.93 (2019), 12 pp.
Alexander Lazar and Michelle L. Wachs
On the homogenized Linial arrangement: intersection lattice and Genocchi numbers
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.
Received: November 15, 2018.
Accepted: February 17, 2019.
Final version: April 1, 2019.
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