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

TL;DR

This paper introduces a new tree-based statistic that interprets the coefficients of the characteristic polynomial of braid deformations, linking combinatorial structures with algebraic properties of hyperplane arrangements.

Contribution

It defines a novel statistic on trees associated with hyperplane arrangement deformations, providing a combinatorial interpretation of characteristic polynomial coefficients.

Findings

The statistic applies to well-known arrangements like Catalan, Shi, Linial, and semiorder.

It establishes a one-to-one correspondence between regions and labeled trees.

The distribution of the statistic matches the coefficients of the characteristic polynomial.

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.