高斯多项式的置乱Vandermonde卷积
Scrambled Vandermonde Convolutions of Gaussian Polynomials
摘要：
众所周知，高斯多项式（即，$q$ -二项式）描述矩形网格中单调路径上的$area$统计量的分布。我们引入了两个新的统计量，$corners$和$cindex$ ;附加“干扰”到网格;并重新评估这些统计量，以证明所有加扰版本的$cindex$统计量与$area$是均匀分布的。我们的主要结果是一个双统计$(cindex,corners)$的生成函数表示为一个两个变量的Vandermonde卷积的原始高斯多项式。证明依赖于不同方向路径之间的显式双射。
英文摘要：
It is well known that Gaussian polynomials (i.e., $q$-binomials) describe the distribution of the $area$ statistic on monotone paths in a rectangular grid. We introduce two new statistics, $corners$ and $cindex$; attach ``ornaments'' to the grid; and re-evaluate these statistics, in order to argue that all scrambled versions of the $cindex$ statistic are equidistributed with $area$. Our main result is a representation of the generating function for the bi-statistic $(cindex,corners)$ as a two-variable Vandermonde convolution of the original Gaussian polynomial. The proof relies on explicit bijections between differently ornated paths.
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