On the Homogenized Linial Arrangement: Intersection Lattice and Genocchi
Numbers
A. Lazar, and M. Wachs. (2018)cite arxiv:1811.06882Comment: 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.
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.
Description
On the Homogenized Linial Arrangement: Intersection Lattice and Genocchi Numbers
cite arxiv:1811.06882Comment: 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
%0 Generic
%1 lazar2018homogenized
%A Lazar, Alexander
%A Wachs, Michelle L.
%D 2018
%K intersection lattice order
%T On the Homogenized Linial Arrangement: Intersection Lattice and Genocchi
Numbers
%U http://arxiv.org/abs/1811.06882
%X 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.
@misc{lazar2018homogenized,
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 M\"obius
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 M\"obius 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.},
added-at = {2019-12-17T18:23:44.000+0100},
author = {Lazar, Alexander and Wachs, Michelle L.},
biburl = {https://www.bibsonomy.org/bibtex/22a2e9bf532cdbe75848ae9b29b5ec3e4/tomhanika},
description = {On the Homogenized Linial Arrangement: Intersection Lattice and Genocchi Numbers},
interhash = {53dde768ab2a8f93acf9fabd13001a93},
intrahash = {2a2e9bf532cdbe75848ae9b29b5ec3e4},
keywords = {intersection lattice order},
note = {cite arxiv:1811.06882Comment: 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},
timestamp = {2019-12-17T18:23:44.000+0100},
title = {On the Homogenized Linial Arrangement: Intersection Lattice and Genocchi
Numbers},
url = {http://arxiv.org/abs/1811.06882},
year = 2018
}