Symmetries of Plane Partitions and the Permanent - Determinant Method
Symmetries of Plane Partitions and the Permanent - Determinant Method
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平面分分的对称性和永久行列式方法
DOI：
10.1016/0097-3165(94)90094-9
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发表时间：
1994
期刊：
影响因子：
--
通讯作者：
G. Kuperberg
中科院分区：
文献类型：
--
作者：
G. Kuperberg
R. P. Stanley（1986，J. Combin.理论系列A43，103-113）给出了10个对称类中每个对称类的平面分区数的公式。本文结合G. Andrews（J. Combin. Theory Ser. A，to appear）和J. Stembridge（The enumeration of totally symmetric plane partitions，preprint）完成了证明所有10个公式的项目。我们列举循环对称，自互补的平面分区。我们首先将平面划分转化为平面上六边形的菱形拼接，或者等价地转化为平面图中的匹配。然后，我们可以使用不变行列式方法或其变体--哈夫尼-普夫朗方法，在这10种情况下，得到矩阵的行列式或普夫朗形式的答案。我们行减少所产生的矩阵在考虑的情况下，以证明公式。类似的行减少过程可以在许多其他情况下进行，我们分析了其他三个对称类的平面分区进行比较。
R. P. Stanley (1986,J. Combin. Theory Ser. A43, 103–113) gives formulas for the number of plane partitions in each of 10 symmetry classes. This paper together with papers of G. Andrews (J. Combin. Theory Ser. A, to appear) and J. Stembridge (The enumeration of totally symmetric plane partitions, preprint) completes the project of proving all 10 formulas. We enumerate cyclically symmetric, self—complementary plane partitions. We first convert plane partitions to tilings of a hexagon in the plane by rhombuses, or equivalently to matchings in a certain planar graph. We can then use the permanent—determinant method or a variant, the Hafnian-Pfaffian method, to obtain the answer as the determinant or Pfaffian of a matrix in each of the 10 cases. We row-reduce the resulting matrix in the case under consideration to prove the formula. A similar row-reduction process can be carried out in many of the other cases, and we analyze three other symmetry classes of plane partitions for comparison.