A branch statistic for trees: Interpreting coefficients of the characteristic polynomial of braid deformations

Abstract

A hyperplane arrangement in ${\mathbb{R}}^n$ is a finite collection of affine hyperplanes. The regions are the connected components of the complement of these hyperplanes. By a theorem of Zaslavsky, the number of regions of a hyperplane arrangement is the sum of coefficients of its characteristic polynomial. Arrangements that contain hyperplanes parallel to subspaces whose defining equations are $x_i-x_j=0$ form an important class called the deformations of the braid arrangement. In a recent work, Bernardi showed that 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 well-studied families like extended Catalan, Shi, Linial and semiorder.

Section 4 restructured, other minor changes, final version

References in corpus (1)

• Computing the Tutte polynomial of a hyperplane arrangement

Cited by in corpus (1)

• Refinements of the braid arrangement and two parameter Fuss-Catalan numbers