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) ^⌊n/2⌋ divides ∑π∈Snqaltmaj(π), where S_n is the set of all permutations of { 1 , 2 , … , n} and altmaj (π) is the alternating major index of π. This leads us to discover a q-analog of n! = 2 ^ℓm, m odd, using the statistic of alternating major index. Moreover, we study the γ-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.