王者体育

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科学研究
报告题目:

王者体育positivity and divisibility of alternating descent polynomials

报告人:

linzhicong jiaoshou(shandongdaxue)

报告时间:

报告地点:

tengxunhuiyiid : 144 157 351

报告摘要:

王者体育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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