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In this paper we consider the hyperplane arrangement in $mathbb{R}^n$ whose hyperplanes are ${x_i + x_j = 1mid 1leq i < jleq n}cup {x_i=0,1mid 1leq ileq n}$. We call it the emph{boxed threshold arrangement} since we show that the bounded regions of this arrangement are contained in an $n$-cube and are in one-to-one correspondence with the labeled threshold graphs on $n$ vertices. The problem of counting regions of this arrangement was studied earlier by Joungmin Song. He determined the characteristic polynomial of this arrangement by relating its coefficients to the count of certain graphs. Here, we provide bijective arguments to determine the number of regions. In particular, we construct certain signed partitions of the set ${-n,dots, n}setminus{0}$ and also construct colored threshold graphs on $n$ vertices and show that both these objects are in bijection with the regions of the boxed threshold arrangement. We independently count these objects and provide closed form formula for the number of regions.
In this paper, we give a basis for the derivation module of the cone over the Shi arrangement of the type $D_ell$ explicitly.
Arrangement theory plays an essential role in the study of the unfolding model used in many fields. This paper describes how arrangement theory can be usefully employed in solving the problems of counting (i) the number of admissible rankings in an u
For all positive integers $t$ exceeding one, a matroid has the cyclic $(t-1,t)$-property if its ground set has a cyclic ordering $sigma$ such that every set of $t-1$ consecutive elements in $sigma$ is contained in a $t$-element circuit and $t$-elemen
The Shi arrangement is an affine arrangement of hyperplanes consisting of the hyperplanes of the Weyl arrangement and their parallel translations. It was introduced by J.-Y. Shi in the study of the Kazhdan-Lusztig representation of the affine Weyl gr
For graphs $G$ and $H$, let $G {displaystylesmash{begin{subarray}{c} hbox{$tinyrm rb$} longrightarrow hbox{$tinyrm p$} end{subarray}}}H$ denote the property that for every proper edge-colouring of $G$ there is a rainbow $H$ in $G$. It is known that