Michelle Wachs: On the homogenized Linial arrangement and Genocchi numbers

Abstract: The homogenized Linial arrangement is a hyperplane
arrangement recently introduced by Gabor Hetyei, who showed that its
number of regions 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 in terms of Dumont-like
permutations. 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. Our techniques also yield type B analogs of
these results, and Dowling arrangement generalizations. As a byproduct
of our work, we obtain new models for the Genocchi and median Genocchi
numbers. This is joint work with Alex Lazar.