论文标题
一个单峰序列，模式为四分之一长
A unimodal sequence with mode at a quarter length
论文作者
论文摘要
我们表明，具有$ m $偶数零件和最大的挂钩长度$ n $的数字$ a（n，m）$，对于$ n \ ge 6 $，具有强烈的单峰[（n-1）/4]。我们通过诱导建立了这一结果，使用林，Xiong和Yan引起的$ 5 $ - 学期复发，以及Zeilberger的算法获得的两次$ 4 $ - 学期复发。序列$ a（n，m）$不是log-concave。使用Möbius变换和交错零的方法，我们获得每个生成函数的每个零$ \ sum_m a（n，m）z^m $都位于圆的左侧一半| z-1 | = 2。此外，作为Wang和Zhang对多项式序列的根几何形状的表征，该序列满足了类型$（1,1）$的复发性，我们看到所有这些零在半个圆圈上密集分布。
We show that the number $A(n,m)$ of partitions with $m$ even parts and largest hook length $n$ is strongly unimodal with mode [(n-1)/4] for $n\ge 6$. We establish this result by induction, using a $5$-term recurrence due to Lin, Xiong and Yan, and two $4$-term recurrences obtained by Zeilberger's algorithm. The sequence $A(n,m)$ is not log-concave. Using Möbius transformation and the method of interlacing zeros, we obtain that every zero of every generating function $\sum_m A(n,m)z^m$ lies on the left half part of the circle |z-1|=2. Moreover, as a direct application of Wang and Zhang's characterization of root geometry of polynomial sequences that satisfy a recurrence of type $(1,1)$, we see that all these zeros are densely distributed on the half circle.
