Worpitzky-compatible subarrangements of braid arrangements and cocomparability graphs

TL;DR

This paper characterizes Worpitzky-compatible subarrangements of braid arrangements using cocomparability graphs and derives new formulas for related Eulerian polynomials, linking combinatorial structures with algebraic properties.

Contribution

It establishes a characterization of Worpitzky-compatible graphic arrangements via cocomparability graphs and provides new formulas for their Eulerian polynomials.

Findings

Worpitzky-compatible graphic arrangements correspond to cocomparability graphs

New formulas for chromatic and graphic Eulerian polynomials of cocomparability graphs

Connection between algebraic arrangements and graph-theoretic structures

Abstract

The class of Worpitzky-compatible subarrangements of a Weyl arrangement together with an associated Eulerian polynomial was recently introduced by Ashraf, Yoshinaga and the first author, which brings the characteristic and Ehrhart quasi-polynomials into one formula. The subarrangements of the braid arrangement, the Weyl arrangement of type $A$, are known as the graphic arrangements. We prove that the Worpitzky-compatible graphic arrangements are characterized by cocomparability graphs. Our main result yields new formulas for the chromatic and graphic Eulerian polynomials of cocomparability graphs.