On the Homogenized Linial Arrangement: Intersection Lattice and Genocchi Numbers

Alexander Lazar, Michelle L. Wachs

Research output: Contribution to conferencePaperpeer-review

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.

Original languageEnglish (US)
StatePublished - 2019
Event31st International Conference on Formal Power Series and Algebraic Combinatorics, FPSAC 2019 - Ljubljana, Slovenia
Duration: Jul 1 2019Jul 5 2019

Conference

Conference31st International Conference on Formal Power Series and Algebraic Combinatorics, FPSAC 2019
CountrySlovenia
CityLjubljana
Period7/1/197/5/19

Keywords

  • Dowling lattices
  • Genocchi numbers
  • Hyperplane arrangements

ASJC Scopus subject areas

  • Algebra and Number Theory

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