论文标题
交替的下降多项式的阳性和分裂性
Positivity and divisibility of alternating descent polynomials
论文作者
论文摘要
Chebikin引入了对置换的交替下降统计量，作为下降统计量的变体。我们表明，置换上的交替下降多项式是通过五届复发关系单峰的。我们还发现了交替的下降多项式交替指数$ q $ analog的二次递归。作为此二次递归的有趣应用，我们表明$（1+q）^{\ lfloor n/2 \ rfloor} $划分$ \ sum_ {π\ in \ in \ mathfrak {s}} _n} _n} q^{q^{\ rm {altmaj} $， $ \ {1,2，\ ldots，n \} $和$ \ rm {altmaj}（π）$的排列是$π$的交替主要索引。这使我们使用交替的主要索引的统计数据，发现$ n！= 2^{\ ell} m $，$ m $ odd的$ q $ -Analog。此外，我们通过使用这两个递归和$ {\ textbf {cd}} $ - index来研究交替下降多项式的$γ$ - 向量。制定了进一步的吸引人的猜想，这表明交替的下降统计量应该得到更多的工作。
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 $\sum_{π\in\mathfrak{S}_n}q^{\rm{altmaj}(π)}$, where $\mathfrak{S}_n$ is the set of all permutations of $\{1,2,\ldots,n\}$ and $\rm{altmaj}(π)$ is the alternating major index of $π$. 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 $γ$-vectors of the alternating descent polynomials by using these two recursions and the ${\textbf{cd}}$-index. Further intriguing conjectures are formulated, which indicate that the alternating descent statistic deserves more work.
