  We study the intersection lattice of a hyperplane arrangement recently
introduced by Hetyei who showed that the number of regions of the arrangement
is a median Genocchi number. Using a different method, we refine Hetyei's
result by providing a combinatorial interpretation of the coefficients of the
characteristic polynomial of the intersection lattice of this arrangement. We
also show that the M\"obius invariant of the intersection lattice is a
(nonmedian) Genocchi number. The Genocchi numbers count a class of permutations
known as Dummont permutations and the median Genocchi numbers count the
derangements in this class. We show that the signless coefficients of the
characteristic polynomial count Dumont-like permutations with a given number of
cycles. This enables us to derive formulas for the generating function of the
characteristic polynomial, which reduce to known formulas for the generating
functions of the Genocchi numbers and the median Genocchi numbers. As a
byproduct of our work, we obtain new models for the Genocchi and median
Genocchi numbers.
