The Homogenized Linial Arrangement and Genocchi Numbers

TL;DR

This paper explores the intersection lattice of a hyperplane arrangement linked to Genocchi numbers, providing combinatorial interpretations and new models that connect these numbers to permutation classes and polynomial coefficients.

Contribution

It offers a combinatorial interpretation of the characteristic polynomial coefficients and introduces new models for Genocchi and median Genocchi numbers, refining previous results.

Findings

Coefficients count Dumont-like permutations with a specified number of cycles

Derived formulas for the generating functions of the characteristic polynomial

Connected the intersection lattice invariants to known Genocchi number formulas

Abstract

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.