• 科學研究
    報告題目:

    Positivity and divisibility of alternating descent polynomials

    報告人:

    林志聰 教授(山東大學)

    報告時間:

    報告地點:

    騰訊會議ID :144 157 351

    報告摘要:

    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.

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