The Eulerian polynomials of type D have only real roots

Abstract

We give an intrinsic proof of a conjecture of Brenti that all the roots of the Eulerian polynomial of type D are real and a proof of a conjecture of Dilks, Petersen, and Stembridge that all the roots of the affine Eulerian polynomial of type B are real, as well. © 2013 Discrete Mathematics and Theoretical Computer Science (DMTCS).