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 characterisitic 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.
 Publication:

arXiv eprints
 Pub Date:
 November 2018
 arXiv:
 arXiv:1811.06882
 Bibcode:
 2018arXiv181106882L
 Keywords:

 Mathematics  Combinatorics;
 52C35 (Primary);
 05A05;
 05A15;
 05B35;
 06A07;
 11B68 (Secondary)
 EPrint:
 12 pages, 4 figures. An extended abstract, accepted to conference proceedings of Formal Power Series and Algebraic Combinatorics (FPSAC), 2019. (V2): Improvements of some results, and minor corrections. (V3): Addition of Theorem 4.5, addition of two references, and minor edits