Journal of Integer Sequences, Vol. 24 (2021), Article 21.5.7

Counting Regions of the Boxed Threshold Arrangement

Priyavrat Deshpande, Krishna Menon, and Anurag Singh
Chennai Mathematical Institute
H1, SIPCOT IT Park, Siruseri
Tamil Nadu 603103


In this paper we consider the hyperplane arrangement in Rn whose hyperplanes are {xi + xj = 1 ∣ 1 ≤ i < jn} ∪ {xi = 0,1 ∣ 1 ≤ in}. 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.

Full version:  pdf,    dvi,    ps,    latex    

(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.

Return to Journal of Integer Sequences home page