Recurrence relations for binomial-Eulerian polynomials

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.

Binomial Eulerian polynomials for colored permutations

“…first appeared in work of Postnikov, Reiner and Williams [24,Section 10.4] on the face enumeration of generalized permutohedra and was further studied in [29] (where the name binomial Eulerian polynomial was adopted) and in [21]. The following statement, which can be derived from a more general result [24,Theorem 11.6] on the h-polynomials of chordal nestohedra, shows that A n (t) shares some of the main properties of the Eulerian polynomial A n (t).…”

Real-rootedness of variations of Eulerian polynomials

“…first studied by Postnikov, Reiner, and Williams [24,Section 10.4], are also γ-positive and have attracted a lot of interest recently [26,23,5,22]. Shareshian and Wachs [26] called them binomial Eulerian polynomials and further studied a symmetric function generalization of them, which are shown to be equivariant γ-positive.…”

“…Another common property of A n (z) and d n (z) is that they both have only real roots, proved by Frobenius [15] and Zhang [34], respectively. It is natural to ask whether A n (z) is real-rooted as well, which was conjectured by Ma, Ma, and Yeh [23] based on empirical evidence. Eulerian polynomials and derangement polynomials can be generalized to s-inversion sequences.…”