A doubly-refined enumeration of alternating sign matrices and descending plane partitions
A doubly-refined enumeration of alternating sign matrices and descending plane partitions
复制标题
交替符号矩阵和降平面划分的双重细化枚举
DOI：
10.1016/j.jcta.2012.09.004
复制
发表时间：
2012
期刊：
影响因子：
--
通讯作者：
P. Zinn
中科院分区：
文献类型：
--
作者：
R. Behrend;P. Francesco;P. Zinn
最近作者证明了，对于任意n，n×n交替符号矩阵(ASM)和降平面划分(DPP)上的某些统计量三元组的分布相等，每个部分至多n.一个ASM A的统计量是A中的广义逆个数，A中的−1ʼS个数和A中第一行1左边的0ʼS个数.DPP D的相应统计量是D中的非特殊部分个数，D中特殊部分的个数和D中nʼS的个数.结果被推广为包括每种类型对象的第四个统计量，其中这是在ASM的最后一行中1右边的0ʼS的数目，以及(n−1)ʼS加上DPP中长度为n−1的行的数目。这一推广是用三统计量母函数的已知等式，以及用三统计量母函数的对应关系来表示每个四统计量母函数的关系来证明的。这些关系是通过将Desnanot-Jacobi恒等式应用于生成函数的行列式表达式来获得的，其中行列式来自标准方法，所述标准方法包括ASM的六顶点模型和DPP的不相交的晶格路径。
It was shown recently by the authors that, for any n, there is equality between the distributions of certain triplets of statistics on n× n alternating sign matrices (ASMs) and descending plane partitions (DPPs) with each part at most n. The statistics for an ASM A are the number of generalized inversions in A, the number of− 1ʼs in A and the number of 0ʼs to the left of the 1 in the first row of A, and the respective statistics for a DPP D are the number of nonspecial parts in D, the number of special parts in D and the number of nʼs in D. Here, the result is generalized to include a fourth statistic for each type of object, where this is the number of 0ʼs to the right of the 1 in the last row of an ASM, and the number of (n− 1) ʼs plus the number of rows of length n− 1 in a DPP. This generalization is proved using the known equality of the three-statistic generating functions, together with relations which express each four-statistic generating function in terms of its three-statistic counterpart. These relations are obtained by applying the Desnanot–Jacobi identity to determinantal expressions for the generating functions, where the determinants arise from standard methods involving the six-vertex model with domain-wall boundary conditions for ASMs, and nonintersecting lattice paths for DPPs.