Coefficients of the Inflated Eulerian Polynomial

TL;DR

This paper characterizes sequences for which inflated Eulerian polynomials exhibit a specific coefficient property, proves their unimodality for all positive sequences, and explores conditions for palindromicity, extending known results in permutation statistics.

Contribution

It provides a complete characterization of sequences satisfying a coefficient coincidence property and proves the unimodality of inflated s-Eulerian polynomials for all positive sequences.

Findings

Sequences with nondecreasing order satisfy the coefficient property.

Inflated s-Eulerian polynomials are unimodal for all positive sequences.

The paper determines conditions under which these polynomials are palindromic.

Abstract

It follows from work of Chung and Graham that for a certain family of polynomials $T_{n}(x)$, derived from the descent statistic on permutations, the coefficient sequence of $T_{n-1}(x)$ coincides with that of the polynomial $T_{n}(x)/\left(1+x+\cdots+x^{n-1}\right)$. We observed computationally that the inflated $\mathbf{s}$-Eulerian polynomial $Q_{n}^{(\mathbf{s})}(x)$, which satisfies $Q_{n}^{(\mathbf{s})}(x) = T_{n}(x)$ when $\mathbf{s}=(1,2,\ldots,n)$, also satisfies this property for many sequences $\mathbf{s}$. In this work we characterize those sequences $\mathbf{s}$ for which the coefficient sequence of $Q_{n-1}^{(\mathbf{s})}(x)$ coincides with that of the polynomial $Q_{n}^{(\mathbf{s})}(x)/\left(1+x+\cdots+x^{s_{n}-1}\right)$. In particular, we show that all nondecreasing sequences satisfy this property. We also settle a conjecture of Pensyl and Savage by showing that the inflated $\mathbf{s}$-Eulerian polynomials are unimodal for all choices of positive integer sequences ${\bf s}$. In addition, we determine when these polynomials are palindromic and show our characterization is equivalent to another of Beck, Braun, K\"oppe, Savage, and Zafeirakopoulos.