  The alternating descent statistic on permutations was introduced by Chebikin
as a variant of the descent statistic. We show that the alternating descent
polynomials on permutations are unimodal via a five-term recurrence relation.
We also found a quadratic recursion for the alternating major index $q$-analog
of the alternating descent polynomials. As an interesting application of this
quadratic recursion, we show that $(1+q)^{\lfloor n/2\rfloor}$ divides
$\sum_{\pi\in\mathfrak{S}_n}q^{\rm{altmaj}(\pi)}$, where $\mathfrak{S}_n$ is
the set of all permutations of $\{1,2,\ldots,n\}$ and $\rm{altmaj}(\pi)$ is the
alternating major index of $\pi$. This leads us to discover a $q$-analog of
$n!=2^{\ell}m$, $m$ odd, using the statistic of alternating major index.
Moreover, we study the $\gamma$-vectors of the alternating descent polynomials
by using these two recursions and the ${\textbf{cd}}$-index. Further intriguing
conjectures are formulated, which indicate that the alternating descent
statistic deserves more work.
