Positivity and divisibility of alternating descent polynomials

Abstract

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 ${\sum }_{\pi \in {\mathfrak{S}}_n}{q}^{\mathrm{altmaj}(\pi )}$, where ${\mathfrak{S}}_n$ is the set of all permutations of $\{1,2,\dots ,n\}$ 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 descent polynomials by using these two recursions and the $\text{cd}$-index. Further intriguing conjectures are formulated, which indicate that the alternating descent statistic deserves more work.

21 pages

Cited by in corpus (1)

• Stability of combinatorial polynomials and its applications