论文标题
麦克马洪关于高维分区的统计数据
MacMahon's statistics on higher-dimensional partitions
论文作者
论文摘要
我们研究了高维分区的一些组合性能，这些特性概括了平面分区。我们介绍了$ d $维分区与非负整数阵列之间的自然培养。该培训具有许多重要的应用。我们在$ d $维分区（称为Corner-Hook卷）上介绍了一个统计数据，其生成功能具有Macmahon的猜想的公式。我们获得了多变量公式，其专业化给出了以平面分区已知的各种公式的类似程序。我们还引入了双重粒度多项式的较高维度类似物，这些双项式多项式是准对称函数，其专门列举了给定形状的高维分区。最后，我们以$ \ mathbb {z}^d $中的定向最后一段渗透模型显示了概率连接。
We study some combinatorial properties of higher-dimensional partitions which generalize plane partitions. We present a natural bijection between $d$-dimensional partitions and $d$-dimensional arrays of nonnegative integers. This bijection has a number of important applications. We introduce a statistic on $d$-dimensional partitions, called the corner-hook volume, whose generating function has the formula of MacMahon's conjecture. We obtain multivariable formulas whose specializations give analogues of various formulas known for plane partitions. We also introduce higher-dimensional analogues of dual Grothendieck polynomials which are quasisymmetric functions and whose specializations enumerate higher-dimensional partitions of a given shape. Finally, we show probabilistic connections with a directed last passage percolation model in $\mathbb{Z}^d$.
