Related papers: Distinguishing $\Bbbk$-configurations

An $(n_k)$-configuration is a set of $n$ points and $n$ lines in the projective plane such that their point-line incidence graph is $k$-regular. The configuration is geometric, topological, or combinatorial depending on whether lines are…

This paper is a sequel to the paper \cite{refGH}. We relate the matroid notion of a combinatorial geometry to a generalization which we call a configuration type. Configuration types arise when one classifies the Hilbert functions and…

A $\Bbbk$-configuration of type $(d_1,\dots,d_s)$ is a specific set of points in $\mathbb P^2$ that has a number of algebraic and geometric properties. For example, the graded Betti numbers and Hilbert functions of all…

An $(n_k)$ configuration is a set of $n$ points and $n$ lines such that each point lies on $k$ lines while each line contains $k$ points. The configuration is geometric, topological, or combinatorial depending on whether lines are…

This paper begins by extending the notion of a combinatorial configuration of points and lines to a combinatorial configuration of points and planes that we refer to as configurations of order $2$. We then proceed to investigate a further…

A recent paper by the first and third authors together with Sabourin raised the question of what the possible Hilbert functions are for fat point subschemes of the form $2p_1+...+2p_r$, for all possible choices of $r$ distinct points in the…

Given five positive integers $v, m,k,\lambda$ and $t$ where $v \geq k \geq t$ and $v \geq m \geq t,$ a $t$-$(v,k,m,\lambda)$ general covering design is a pair $(X,\mathcal{B})$ where $X$ is a set of $v$ elements (called points) and…

To any finite ordered subset and any finite partition of a group a set of tuples of positive integers, named as configurations, is associated that describes the group's behavior. The present paper provides an exposition of this notion and…

Given a collection of $l$ general lines $\ell_1,\ldots,\ell_{l}$ in $\pr^2$, the star configuration $\XX(l)$ is the set of points constructed from all pairwise intersections of these lines. For each non-negative integer $d$, we compute the…

In a series of papers and in his 2009 book on configurations Branko Gr\"unbaum described a sequence of operations to produce new $(n_{4})$ configurations from various input configurations. These operations were later called the "Gr\"unbaum…

Suppose $k$ is a positive integer and $\mathcal{X}$ is a $k$-fold packing of the plane by infinitely many arc-connected compact sets, which means that every point of the plane belongs to at most $k$ sets. Suppose there is a function…

Presented is a wonderful compactification of n distinct labeled points in X away from D, where X is a nonsingular variety and D is a nonsingular proper subvariety. When D is empty, it is the Fulton-MacPherson configuration space.

This paper introduces complexes of linear varieties, called inclics (for INductively Constructible LInear ComplexeS). As examples, we study galaxies (these are constructed starting with a star configuration to which we add general points in…

In this note, we consider ordered partitions of integers such that each entry is no more than a fixed portion of the sum. We give a method for constructing all such compositions as well as both an explicit formula and a generating function…

For every integer $k \geq 2$ and every $A \subseteq \mathbb{N}$, we define the \emph{$k$-directions sets} of $A$ as $D^k(A) := \{{\bf a} / \|{\bf a}\| : {\bf a} \in A^k\}$ and $D^{\underline{k}}(A) := \{{\bf a} / \|{\bf a}\| : {\bf a} \in…

An integer composition of a nonnegative integer $n$ is a tuple $(\pi_1,\ldots,\pi_k)$ of nonnegative integers whose sum is $n$; the $\pi_i$'s are called the parts of the composition. For fixed number $k$ of parts, the number of $f$-weighted…

Lineability is a property enjoyed by some subsets within a vector space X. A subset A of X is called lineable whenever A contains, except for zero, an infinite dimensional vector subspace. If, additionally, X is endowed with richer…

The $ k $-configuration space $ B_k\Gamma $ of a topological space $ \Gamma $ is the space of sets of $ k $ distinct points in $ \Gamma $. In this paper, we consider the case where $ \Gamma $ is a graph of circumference at most $1$. We show…

A $k$-regular graph is called a divisible design graph (DDG for short) if its vertex set can be partitioned into $m$ classes of size $n$, such that two distinct vertices from the same class have exactly $\lambda_1$ common neighbors, and two…

The two basic equations satisfied by the parameters of a block design define a three-dimensional affine variety $\mathcal{D}$ in $\mathbb{R}^{5}$. A point of $\mathcal{D}$ that is not in some sense trivial lies on four lines lying in…