The Hillman-Grassl Correspondence and the Enumeration of Reverse Plane Partitions
The Hillman-Grassl Correspondence and the Enumeration of Reverse Plane Partitions
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Hillman-Grassl对应和反向平面划分的枚举
DOI：
10.1016/0097-3165(81)90041-8
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发表时间：
1981
期刊：
影响因子：
--
通讯作者：
E. Gansner
中科院分区：
文献类型：
--
作者：
E. Gansner
Hillman和Grassl设计了反向平面分区和相同形状的非负整数阵列之间的对应关系，使他们能够轻松地枚举反向平面分区，并提供了钩长度和平面分区之间的组合连接。在这项工作中，这个对应的属性的集合，包括两个特征，这张地图熟悉的Schensted-Knuth对应。利用这些性质导出了逆平面分割和对称逆平面分割的母函数关于沿对角线的和沿着的简单表达式。同样一般的结果得到移位反向平面分区使用一种新类型的挂钩，从而证明了斯坦利的猜想。
Hillman and Grassl have devised a correspondence between reverse plane partitions and nonnegative integer arrays of the same shape that allowed them to easily enumerate reverse plane partitions and provided a combinatorial connection between hook lengths and plane partitions. In this work, a collection of properties of this correspondence are presented, including two characterizations that relate this map to the familiar Schensted-Knuth correspondence. These properties are used to derive simple expressions for the generating functions of reverse plane partitions and symmetric reverse plane partitions with respect to sums along the diagonals. Equally general results are obtained for shifted reverse plane partitions using a new type of hook, thereby proving a conjecture of Stanley.