Séminaire Lotharingien de Combinatoire, 86B.23 (2022), 12 pp.
Priyavrat Deshpande and Krishna Menon
A Statistic for Regions of Braid Deformations
Abstract.
An arrangement of hyperplanes in Rn is a finite collection of hyperplanes.
The regions are the connected components of the complement of the union of these hyperplanes.
By a theorem of Zaslavsky, the number of regions of a hyperplane arrangement is the sum of the absolute values of the coefficients of its characteristic polynomial.
Arrangements that contain hyperplanes parallel to subspaces whose defining equations are xi - xj = 0 form an important class called the deformations of the braid arrangement.
In a recent work, Bernardi showed regions of certain deformations are in one-to-one correspondence with certain labeled trees.
In this article, we define a statistic on these trees such that the distribution is given by the coefficients of the characteristic polynomial.
In particular, our statistic applies to the well-studied families like extended Catalan, Shi, Linial and semiorder.
Received: November 25, 2021.
Accepted: March 4, 2022.
Final version: April 1, 2022.
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