Recurrence relations for binomial-Eulerian polynomials

TL;DR

This paper investigates properties of binomial-Eulerian polynomials, deriving recurrence relations and generating functions, and provides combinatorial interpretations related to hyperplane arrangements.

Contribution

It offers three constructive proofs of recurrence relations and new combinatorial insights into the Betti numbers of hyperplane arrangement complements.

Findings

Derived recurrence relations for binomial-Eulerian polynomials

Established generating functions for these polynomials

Provided combinatorial interpretation of Betti numbers

Abstract

Binomial-Eulerian polynomials were introduced by Postnikov, Reiner and Williams. In this paper, properties of the binomial-Eulerian polynomials, including recurrence relations and generating functions are studied. We present three constructive proofs of the recurrence relations for binomial-Eulerian polynomials. Moreover, we give a combinatorial interpretation of the Betti number of the complement of the k-equal real hyperplane arrangement.