Coefficients of the Inflated Eulerian Polynomial

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 $\mathbf{s}=(1,2,\dots ,n)$, 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 $\mathbf{s}$. In addition, we determine when these polynomials are palindromic and show our characterization is equivalent to another of Beck, Braun, Köppe, Savage, and Zafeirakopoulos.

New results and new collaborator added. 26 pages, 4 figures