论文标题
飞机上点配置的双孔酮细分
Bimonotone Subdivisions of Point Configurations in the Plane
论文作者
论文摘要
两维的双孔酮细分是分区，其各个方面都是垂直的或具有非负斜率。它们对应于强度正相关的随机变量的概率分布的统计估计。与点配置的子分数总数相比，双膜酮细分的数量为随机变量积极依赖的频率提供了见解。我们提供递归以及双膜酮数量和$ 2 \ times n $网格配置的总分支的公式。此外，我们将前者连接到较大的Schröder数字。我们还表明，$ 2 \ times n $网格的双孔酮数量和总细分数在渐近上相等。然后，我们提供用于计算任何$ m \ times n $网格的双膜酮细分的算法。最后，我们证明了$ m \ times n $网格的所有双孔酮三角剖分都通过翻转连接。这引起了计算$ M \ times n $网格的双孔酮（和总）三角剖分的数量的算法。
Bimonotone subdivisions in two dimensions are subdivisions all of whose sides are either vertical or have nonnegative slope. They correspond to statistical estimates of probability distributions of strongly positively dependent random variables. The number of bimonotone subdivisions compared to the total number of subdivisions of a point configuration provides insight into how often the random variables are positively dependent. We give recursions as well as formulas for the numbers of bimonotone and total subdivisions of $2\times n$ grid configurations in the plane. Furthermore, we connect the former to the large Schröder numbers. We also show that the numbers of bimonotone and total subdivisions of a $2\times n$ grid are asymptotically equal. We then provide algorithms for counting bimonotone subdivisions for any $m \times n$ grid. Finally, we prove that all bimonotone triangulations of an $m \times n$ grid are connected by flips. This gives rise to an algorithm for counting the number of bimonotone (and total) triangulations of an $m\times n$ grid.
