[Paper Review] Eulerian polynomials on segmented permutations

Arthur Nunge|arXiv (Cornell University)|May 4, 2018

Advanced Combinatorial Mathematics2 references3 citations

TL;DR

This paper introduces generalized Eulerian polynomials and numbers on segmented permutations, extending classical Eulerian polynomials and ordered Bell polynomials via a descent statistic on permutations with bars. It derives a $q$-analog generating function and establishes a Worpitzky-type identity, unifying combinatorial sequences like Stirling numbers and ordered Bell numbers through noncommutative Hopf algebra structures in segmented composition quasi-symmetric functions.

ABSTRACT

We define a generalization of the Eulerian polynomials and the Eulerian numbers by considering a descent statistic on segmented permutations coming from the study of 2-species exclusion processes and a change of basis in a Hopf algebra. We give some properties satisfied by these generalized Eulerian numbers. We also define a $q$-analog of these Eulerian polynomials which gives back usual Eulerian polynomials and ordered Bell polynomials for specific values of its variables. We also define a noncommutative analog living in the algebra of segmented compositions. It gives us an explicit generating function and some identities satisfied by the generalized Eulerian polynomials such as a Worpitzky-type relation.

Motivation & Objective

• To generalize Eulerian polynomials and numbers by introducing a descent statistic on segmented permutations arising from 2-species exclusion processes.
• To define a $q$-analog of these polynomials that recovers both standard Eulerian polynomials and ordered Bell polynomials at specific parameter values.
• To construct a noncommutative analog in the algebra of segmented compositions, enabling new generating function identities.
• To establish a Worpitzky-type identity for the generalized polynomials, extending classical results to this broader combinatorial framework.

Proposed method

• Define segmented permutations and compositions, introducing descent and segmentation statistics on them.
• Introduce generalized Eulerian numbers $T(n,k)$ and refined counts $K(n,i,j)$ for $i$ descents and $j$ bars.
• Construct the $q$-analog polynomial $\alpha_n(t,q) = \sum_{\sigma \in \mathfrak{P}_n} t^{\operatorname{des}(\sigma)} q^{\operatorname{seg}(\sigma)}$, generalizing $n!$ and ordered Bell numbers.
• Develop a noncommutative analog $\mathcal{A}_n(t,q)$ in the algebra $\mathbf{SCQSym}$ using the ribbon basis $R_I$, with explicit expression in terms of $S^I$.
• Derive the generating function $G(t,q,x) = 1 + \frac{e^{x(1-t)} - 1}{1 + q - (t + q)e^{x(1-t)}}$ via series expansion and algebraic manipulation.
• Use the generating function to prove a generalized Worpitzky identity involving discrete derivatives and coefficients $K(n,i,j)$.

Experimental results

Research questions

• RQ1 How can Eulerian polynomials be generalized to segmented permutations with bar placements and descent statistics?
• RQ2 What is the structure of the $q$-analog $\alpha_n(t,q)$, and how does it unify Eulerian and ordered Bell polynomials?
• RQ3 Can a noncommutative analog of the generalized Eulerian polynomials be constructed in $\mathbf{SCQSym}$, and what is its generating function?
• RQ4 Does a Worpitzky-type identity hold for the generalized Eulerian numbers $K(n,i,j)$, and how does it relate to discrete derivatives of monomials?
• RQ5 What combinatorial properties, such as unimodality, do the generalized Eulerian triangles and tetrahedra exhibit?

Key findings

• The generating function for $\alpha_n(t,q)$ is explicitly given by $G(t,q,x) = 1 + \frac{e^{x(1-t)} - 1}{1 + q - (t + q)e^{x(1-t)}}$, providing a complete analytic description.
• Specializing $q=0$ recovers the classical Eulerian polynomials $P_n(t)$, while $t=0$ yields the ordered Bell polynomials.
• A generalized Worpitzky identity is proven: $\binom{k+r-1}{r}\Delta^{r+1}((k-1)^n) = \sum_{i=0}^{k-1} \binom{n+k-i}{n-1} K(n,i,r)$, extending classical results.
• The noncommutative analog $\mathcal{A}_n(t,q)$ is expressed as $\sum_{I|\!|\!=n} t^{\operatorname{des}(I)} q^{\operatorname{seg}(I)} R_I$, with a closed-form generating function derived via algebraic manipulation.
• The coefficients $K(n,i,j)$ are shown to be unimodal in rows and columns up to $n=1000$, supporting conjectured combinatorial structure.
• The generating function for $\alpha_n(t,q)$ leads to a Dobiński-type identity: $\frac{P_n(t)}{(1-t)^{n+1}} = \sum_{k \geq 0} (1+t)^{k-1} \frac{k^n}{2^{k-1}}$, generalizing known Eulerian identities.