Positivity and divisibility of enumerators of alternating descents

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, called alternating Eulerian polynomials, are unimodal via a five-term recurrence relation. We also find a quadratic recursion for the alternating major index q-analog of the alternating Eulerian polynomials. As an interesting application of this quadratic recursion, we show that $$(1+q)^{\lfloor n/2\rfloor }$$ divides $$\sum _{\pi \in {{\mathfrak {S}}}_n}q^{\mathrm{altmaj}(\pi )}$$ , where $${{\mathfrak {S}}}_n$$ is the set of all permutations of $$\{1,2,\ldots ,n\}$$ and $$\mathrm{altmaj}(\pi )$$ 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 Eulerian polynomials by using these two recursions and the cd-index. Further intriguing conjectures are formulated, which indicate that the alternating descent statistic deserves more work.