论文标题
$ q $ -hook公式的极限法则的度量空间
The metric space of limit laws for $q$-hook formulas
论文作者
论文摘要
在较早的工作中，Billey-Konvalinka- Swanson研究了Stanley的$ Q $ -Hook长度公式的系数的渐近分布，或者同等地是标准的直形状标准tableaux和某些偏斜形状的主要指数。我们将这些调查扩展到Stanley的$ Q $ -HOOK-CONTENT公式，与Semistandard Tableaux和Björner的$ Q $ -Hook长度公式 - 与标记森林的线性扩展相关的障碍。我们表明，尽管它们的系数是'一般'渐近'正常的``一般性''，但仍有许多非正常极限定律。更确切地说，我们在几个制度中介绍并彻底描述了这些统计数据分布的度量空间的紧凑型封闭。额外的极限分布涉及通用统一总和分布，这些分布在某些降低序列空间上以$ 2 $ norm的形式拓扑参数化。这些分布在Lévy度量中的关闭产生了尘土盆地分布的空间。作为一个应用程序，我们完全对盒子中的平面分区上大小统计的限制分布进行了分类。
In earlier work, Billey--Konvalinka--Swanson studied the asymptotic distribution of the coefficients of Stanley's $q$-hook length formula, or equivalently the major index on standard tableaux of straight shape and certain skew shapes. We extend those investigations to Stanley's $q$-hook-content formula related to semistandard tableaux and $q$-hook length formulas of Björner--Wachs related to linear extensions of labeled forests. We show that, while their coefficients are ``generically'' asymptotically normal, there are uncountably many non-normal limit laws. More precisely, we introduce and completely describe the compact closure of the metric space of distributions of these statistics in several regimes. The additional limit distributions involve generalized uniform sum distributions which are topologically parameterized by certain decreasing sequence spaces with bounded $2$-norm. The closure of these distributions in the Lévy metric gives rise to the space of DUSTPAN distributions. As an application, we completely classify the limiting distributions of the size statistic on plane partitions fitting in a box.
