  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.

Number Theory;Combinatorics