Related papers: Accurate Arrangements
Let $(\mathscr{A},m)$ be a free multiarrangement, and let $\mathscr{E}$ be an extension of $(\mathscr{A},m)$. It is well known that if $\mathscr{A}$ is the Coxeter arrangement of type $A_2$, then a free extension of $(\mathscr{A},m)$ always…
We introduce the class of MAT-free hyperplane arrangements which is based on the Multiple Addition Theorem by Abe, Barakat, Cuntz, Hoge, and Terao. We also investigate the closely related class of MAT2-free arrangements based on a recent…
Finite-free additive and multiplicative convolutions are operations on the set of polynomials with real roots, introduced independently by Szeg\"{o} and Walsh in the 1920s. These operations have regained some interest, in the last decade,…
Many important theorems in combinatorics, such as Szemer\'edi's theorem on arithmetic progressions and the Erd\H{o}s-Stone Theorem in extremal graph theory, can be phrased as statements about independent sets in uniform hypergraphs. In…
We prove that every infinite sequence of skew-symmetric or symmetric matrices M_1, M_2, ... over a fixed finite field must have a pair M_i, M_j (i<j) such that M_i is isomorphic to a principal submatrix of the Schur complement of a…
The Ish arrangement was introduced by Armstrong to give a new interpretation of the $q,t$-Catalan numbers of Garsia and Haiman. Armstrong and Rhoades showed that there are some striking similarities between the Shi arrangement and the Ish…
A balanced pair in a finite ordered set $P=(V,\leq)$ is a pair $(x,y)$ of elements of $V$ such that the proportion of linear extensions of $P$ that put $x$ before $y$ is in the real interval $[1/3, 2/3]$. We prove that every finite $N$-free…
We introduce the use of liaison addition to the study of hyperplane arrangements. For an arrangement, $\mathcal A$, of hyperplanes in $\mathbb P^n$, $\mathcal A$ is free if $R/J$ is Cohen-Macaulay, where $J$ is the Jacobian ideal of…
The relation space of a hyperplane arrangement is the vector space of all linear dependencies among the defining forms of the hyperplanes in the arrangement. In this paper, we study the relationship between the relation space and the…
A central arrangement $\A$ of hyperplanes in an $\ell$-dimensional vector space $V$ is said to be {\it totally free} if a multiarrangement $(\A, m)$ is free for any multiplicity $ m : \A\to \Z_{> 0}$. It has been known that $\A$ is totally…
We prove Anzis and Tohaneanu conjecture, that is the Dirac-Motzkin conjecture for supersolvable line arrangements in the projective plane over an arbitrary field of characteristic zero. Moreover, we show that a divisionally free…
Ziegler showed that free arrangements have free restricted multiarrangements (multirestrictions). After Ziegler's work, several results concerning "reverse direction", namely characterizing freeness of an arrangement via that of…
We study the combinatorics of pseudoline arrangements in the real projective plane. Our focus lies on two classes of arrangements: simplicial arrangements and arrangements whose characteristic polynomials have only real roots. We derive…
We study some aspects of divisionally free arrangements which were recently introduced by Abe. Crucially, Terao's conjecture on the combinatorial nature of freeness holds within this class. We show that while it is compatible with products,…
An m-free hyperplane arrangement is a generalization of a free arrangement. Holm asked the following two questions: (1)Does m-free imply (m+1)-free for any arrangement? (2)Are all arrangements m-free for m large enough? In this paper, we…
A saturated D-optimal design is a {+1,-1} square matrix of given order with maximal determinant. We search for saturated D-optimal designs of orders 19 and 37, and find that known matrices due to Smith, Cohn, Orrick and Solomon are optimal.…
In 1989, Ziegler introduced the concept of a multi-arrangement. One natural example is the reflection arrangement of a unitary reflection group with multiplicity given by the number of reflections associated with each hyperplane. For all…
Given an irreducible root system, the Worpitzky-compatible subsets are defined by a geometric property of the alcoves inside the fundamental parallelepiped of the root system. This concept is motivated and mainly understood through a…
The set of weights of a finite-dimensional representation of a reductive Lie algebra has a natural poset structure ("weight poset"). Studying certain combinatorial problems related to antichains in weight posets, we realised that the best…
We introduce and study a digraph analogue of Stanley's $\psi$-graphical arrangements from the perspectives of combinatorics and freeness. Our arrangements form a common generalization of various classes of arrangements in literature…