科学研究
报告题目:

Positivity and divisibility of alternating descent polynomials

报告人:

林志聪 教授(山东大学)

报告时间:

报告地点:

腾讯会议ID :144 157 351

报告摘要:

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 $\sum_{\pi\in\bs_n}q^{\altmaj(\pi)}$, where $\bs_n$ is the set of all permutations of $\{1,2,\ldots,n\}$ and $\altmaj(\pi)$ 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 {\bf cd}-index. Further intriguing conjectures are formulated, which indicate that the alternating descent statistic deserves more work.