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Counting Regions of the Boxed Threshold Arrangement
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Priyavrat Deshpande, Krishna Menon, and Anurag Singh

Chennai Mathematical Institute

H1, SIPCOT IT Park, Siruseri

Tamil Nadu 603103

India

**Abstract:**

In this paper we consider the hyperplane arrangement in **R**^{n}
whose hyperplanes are {*x*_{i}
+ *x*_{j} = 1 ∣
1 ≤ *i* < *j* ≤ *n*} ∪
{*x*_{i} = 0,1 ∣
1 ≤ *i* ≤ *n*}. We call it the *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 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*, ..., *n*}\{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 a closed form formula
for the number of regions.

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(Concerned with sequences
A005840
A039757
A341769.)

Received February 3 2021; revised versions received February 19 2021; February 21 2021.
Published in *Journal of Integer Sequences*,
April 26 2021.

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