Related papers: The Shi arrangements and the Bernoulli polynomials
A characteristic polynomial is an important invariant in the field of hyperplane arrangement. For the Linial arrangement of any irreducible root system, Postnikov and Stanley conjectured that all roots of the characteristic polynomial have…
For a finite Coxeter group, a subword complex is a simplicial complex associated with a pair (Q, \pi), where Q is a word in the alphabet of simple reflections, $\pi$ is a group element. We discuss the transformations of such a complex…
Given an ordered set partition, when one insert a number of bars in-between the blocks of the ordered set partition the result is a barred preferential arrangement. In this study, using the notion of barred preferential arrangements we…
The collection of reflecting hyperplanes of a finite Coxeter group is called a reflection arrangement and it appears in many subareas of combinatorics and representation theory. We focus on the problem of counting regions of reflection…
In this paper, some formulae for Genoochi polynomials of higher order are derived using the fact that sets of Bernoulli and Euler polynomials of higher order form basis for the polynomial space.
We describe a family of hyperplane arrangements depending on a positive integer parameter $r$, which we refer to as the $r$-braid arrangements, and which can be viewed as a generalization of the classical braid arrangement. The wonderful…
We introduce a partition of (coweight) lattice points inside the dilated fundamental parallelepiped into those of partially closed simplices. This partition can be considered as a generalization and a lattice points interpretation of the…
The Tutte polynomial is originally a bivariate polynomial which enumerates the colorings of a graph and of its dual graph. Ardila extended in 2007 the definition of the Tutte polynomial on the real hyperplane arrangements. He particularly…
The purpose of this paper is to derive some identities of the higher order generalized twisted Bernoulli numbers and polynomials attached to $\chi$ from the properties of the p-adic invariant integrals.
In his study of Kazhdan-Lusztig cells in affine type $A$, Shi has introduced an affine analog of Robinson-Schensted correspondence. We generalize the Matrix-Ball Construction of Viennot and Fulton to give a more combinatorial realization of…
Originally in 1954 the Tutte polynomial was a bivariate polynomial associated to a graph in order to enumerate the colorings of this graph and of its dual graph at the same time. However the Tutte polynomial reveals more of the internal…
We define and study the Tutte polynomial of a hyperplane arrangement. We introduce a method for computing it by solving an enumerative problem in a finite field. For specific arrangements, the computation of Tutte polynomials is then…
In [9], Terao proved the freeness of multi-Coxeter arrangements with constant multiplicities by giving an explicit construction of bases. Combining it with algebro-geometric method, Yoshinaga proved the freeness of the extended Catalan and…
In this article, we consider the multiarrangements whose underlying arrangements are the Coxeter arrangement of type $B_2$. For some special multiplicities, we give an explicit description of bases for the derivation modules. As an…
Let $W$ be a finite Weyl group and $\A$ be the corresponding Weyl arrangement. A deformation of $\A$ is an affine arrangement which is obtained by adding to each hyperplane $H\in\A$ several parallel translations of $H$ by the positive root…
We study the exactness of certain combinatorially defined complexes which generalize the Orlik-Solomon algebra of a geometric lattice. The main results pertain to complex reflection arrangements and their restrictions. In particular, we…
This note is a complement of a recent paper about low elements in affine Coxeter groups. We explain in terms of ad-nilpotent ideals of a Borel subalgebra why the minimal elements of dominant Shi regions are low. We also give a survey of the…
The class of Worpitzky-compatible subarrangements of a Weyl arrangement together with an associated Eulerian polynomial was recently introduced by Ashraf, Yoshinaga and the first author, which brings the characteristic and Ehrhart…
The number of flats of a hyperplane arrangement is considered as a generalization of the Bell number and the Stirling number of the second kind. Robert Gill gave the exponential generating function of the number of flats of the extended…
We investigate several hyperplane arrangements that can be viewed as deformations of Coxeter arrangements. In particular, we prove a conjecture of Linial and Stanley that the number of regions of the arrangement x_i - x_j = 1, 1 \leq i<j…