[Paper Review] Recurrence relations for binomial-Eulerian polynomials

Jun Ma, Shi-Mei Ma+1 more|arXiv (Cornell University)|Nov 24, 2017

Advanced Combinatorial Mathematics14 references5 citations

TL;DR

This paper establishes three constructive proofs of recurrence relations for binomial-Eulerian polynomials, providing a bijective solution to the symmetry problem of these polynomials via descent and ascent statistics on a special set of permutations. A key contribution is a combinatorial interpretation of the Betti numbers of the complement of the $k$-equal real hyperplane arrangement using signed cycle statistics on permutations.

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.

Motivation & Objective

• To provide constructive proofs of recurrence relations for binomial-Eulerian polynomials.
• To solve Problem 1.2 by giving a bijective proof of the symmetry of $\widetilde{A}_n(x)$ using descent and ascent statistics on $\mathcal{Q}_n$.
• To establish a combinatorial interpretation of the Betti numbers of the complement of the $k$-equal real hyperplane arrangement.
• To derive generating functions for multivariable binomial-Eulerian polynomials and relate them to known orthogonal polynomials.

Proposed method

• Derive a recurrence for $\widetilde{A}_n(x)$ using exponential generating functions and differential equations.
• Construct three distinct proofs of the recurrence relation for $\widetilde{A}_n(x)$, combining combinatorial and algebraic techniques.
• Introduce a multivariable generating function $\widetilde{A}(x,y,q;z)$ to model signed cycle and excedance statistics on permutations.
• Use the generating function to derive identities involving $\widetilde{A}_n(x,1,-1,-1)$ and $\widetilde{A}_n(x,1,-1,1)$, linking them to Betti numbers.
• Establish a connection between the polynomials $T_n(q)$, defined as sums over cycle statistics on $\widehat{\mathcal{Q}}_{n+1}$, and Charlier polynomials.
• Prove that $T_n(q)$ has only real zeros by relating it to orthogonal Charlier polynomials.

Experimental results

Research questions

• RQ1 Can the symmetry of the binomial-Eulerian polynomial $\widetilde{A}_n(x)$ be proven bijectively using descent and ascent statistics on $\mathcal{Q}_n$?
• RQ2 What is the combinatorial meaning of the Betti numbers $B(n,k)$ of the complement of the $k$-equal real hyperplane arrangement?
• RQ3 How do multivariable generating functions for binomial-Eulerian polynomials encode signed statistics on permutations?
• RQ4 What is the relationship between the cycle-counting polynomials $T_n(q)$ and classical orthogonal polynomials?
• RQ5 Do the polynomials $T_n(q)$, defined via cycle statistics on $\widehat{\mathcal{Q}}_{n+1}$, have only real zeros?

Key findings

• The recurrence relation for $\widetilde{A}_n(x)$ is proven in three distinct constructive ways, with Theorem 2.11 providing a bijective proof of the symmetry of $\widetilde{A}_n(x)$ via descent and ascent statistics on $\mathcal{Q}_n$.
• The Betti number $B(n,k)$ is combinatorially interpreted as the signed sum $\sum_{\sigma \in \widehat{\mathcal{Q}}_{n+1}} (-1)^{{\rm cyc}(\sigma)+{\rm aexc}(\sigma)}$ over permutations with $n-k$ excedances.
• The generating function $\sum_{n \geq 0} B_n(x) \frac{z^n}{n!} = \frac{e^z + x e^{(2+x)z}}{1+x}$ is derived, linking Betti numbers to exponential generating functions.
• The cycle-counting polynomial $T_n(q) = \sum_{\sigma \in \widehat{\mathcal{Q}}_{n+1}} q^{{\rm cyc}(\sigma)}$ satisfies the recurrence $T_{n+1}(q) = (n+1+q)T_n(q) - nT_{n-1}(q)$ with $T_0(q) = q$, $T_1(q) = q + q^2$.
• The polynomial $T_n(q)$ is shown to be equal to $(-1)^n q C_n^{(1)}(-q)$, where $C_n^{(1)}(x)$ is the Charlier polynomial, implying that $T_n(q)$ has only real zeros for all $n \geq 0$.
• The identity $\sum_{\sigma \in \widehat{\mathcal{Q}}_{n+1}} (-1)^{{\rm cyc}(\sigma)} = n-1$ is proven for $n \geq 2$, providing a closed-form evaluation of the alternating cycle sum.