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

Abstract

A. A hyperplane arrangement in 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.

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

“…. This allows us to use the interpretation of the coefficients of χ C (m) n (t) from [6] to describe our required statistic. We first recall this interpretation.…”

“…The second labeled Dyck path in the decorated Dyck path given in Figure4has 2 compartments. If S is a transitive set (see[4, Definition 3.5]) an interpretation of the coefficients of χ C S n (t) is given in[6, Theorem 3.5]. Just as above, this can be used along with Theorem 2.10 to obtain an interpretation for the coefficients of χ A S n (t).…”