Plane Partitions(平面分区)研究综述
Plane Partitions 平面分区 - The Hilbert space is spanned by states labeled by plane partitions, and, writing them as a product of interlaced integer partitions, we define a fermion-boson duality for plane partitions. [1] The methods used are from representation theory and combinatorics; in particular, we make explicit the close connection with MacMahon’s enumeration of plane partitions, and prove a new q-binomial identity in this setting. [2] Then, we calculate the Schur functions of plane partitions of 4, we find that the plane partitions become Young diagrams, and the Schur functions on plane partitions become Schur functions on Young diagrams when h1 = 1, h2 = −1, and h3 = 0. [3] We study certain bijection between plane partitions and $\mathbb{N}$-matrices. [4] For instance, we determine certain corner distributions of random lozenge tilings and plane partitions. [5] One of the most impressive instances of the cyclic sieving phenomenon is a theorem of Rhoades asserting that the set of plane partitions in a rectangular box under the action of promotion exhibits cyclic sieving. [6] The partition hyperplane partitions the current space according to the maximum scatter direction of the majority set in the current space. [7] Procházka and Rapčák, then proposed to interpret Y algebra as a truncation of affine Yangian whose module is directly connected to plane partitions (PP). [8] In their paper about a dual of MacMahon's classical theorem on plane partitions, Ciucu and Krattenthaler proved a closed form product formula for the tiling number of a hexagon with a "shamrock", a union of four adjacent triangles, removed in the center (Proc. [9]希尔伯特空间由平面分区标记的状态跨越，并且，将它们写为交错整数分区的乘积，我们定义了平面分区的费米子-玻色子对偶性。 [1] 使用的方法来自表示论和组合学；特别是，我们明确了与 MacMahon 平面分区枚举的密切联系，并在此设置中证明了一个新的 q-二项式恒等式。 [2] 然后，我们计算4的平面分区的Schur函数，我们发现平面分区变成Young图，当h1 = 1，h2 = -1，h3 = 0时，平面分区上的Schur函数变成Young图上的Schur函数. [3] 我们研究了平面分区和$\mathbb{N}$-矩阵之间的某些双射。 [4] 例如，我们确定随机菱形平铺和平面分区的某些角分布。 [5] 循环筛分现象最令人印象深刻的例子之一是 Rhoades 定理，该定理断言，在促进作用下，矩形框中的平面分区集表现出循环筛分。 [6] 分区超平面根据当前空间中设置的多数派的最大散射方向对当前空间进行分区。 [7] Procházka 和 Rapčák，然后提出将 Y 代数解释为仿射阳根的截断，其模块直接连接到平面分区 (PP)。 [8] 在他们关于平面分区上麦克马洪经典定理的对偶的论文中，Ciucu 和 Krattenthaler 证明了六边形与“三叶草”的平铺数的封闭形式乘积公式，“三叶草”是四个相邻三角形的并集，在中心被删除（Proc. [9]
Reverse Plane Partitions
Sulzgruber's rim hook insertion and the Hillman-Grassl correspondence are two distinct bijections between the reverse plane partitions of a fixed partition shape and multisets of rim-hooks of the same partition shape. [1] Morales, Pak and Panova found two $q$-analogs of Naruse's hook length formula over semistandard Young tableaux (SSYTs) and reverse plane partitions (RPPs). [2] Morales, Pak and Panova found two $q$-analogues of Naruse's formula respectively by counting semistandard Young tableaux of shape $\lambda/\mu$ and reverse plane partitions of shape $\lambda/\mu$. [3]Sulzgruber 的轮辋钩插入和 Hillman-Grassl 对应是固定分区形状的反向平面分区和相同分区形状的多组轮辋钩之间的两个不同的双射。 [1] Morales、Pak 和 Panova 在半标准 Young tableaux (SSYTs) 和反向平面分区 (RPPs) 上发现了 Naruse 的钩子长度公式的两个 $q$-analogs。 [2] Morales、Pak 和 Panova 通过计算形状为 $\lambda/\mu$ 的半标准 Young tableaux 和形状为 $\lambda/\mu$ 的反向平面分区，分别找到了 Naruse 公式的两个 $q$-类似物。 [3]
Descending Plane Partitions
Based on the main result from the first paper, we construct a bijective proof of the enumeration formula for alternating sign matrices and of the fact that alternating sign matrices are equinumerous with descending plane partitions. [1] We evaluate a curious determinant, first mentioned by George Andrews in 1980 in the context of descending plane partitions. [2]基于第一篇论文的主要结果，我们构造了交替符号矩阵的枚举公式的双射证明，以及交替符号矩阵与下降平面分区等数的事实。 [1] 我们评估一个奇怪的行列式，由 George Andrews 于 1980 年在下降平面分区的背景下首次提到。 [2]
Shifted Plane Partitions
We give a product formula for the number of shifted plane partitions of shifted double staircase shape with bounded entries. [1] We show that there is the same number of ( n , l ) -alternating sign trapezoids as there is of column strict shifted plane partitions of class l − 1 with at most n parts in the top row, thereby proving a result that was conjectured independently by Behrend and Aigner. [2]我们给出了有界条目的移位双楼梯形状的移位平面分区数的乘积公式。 [1] 我们证明了 (n , l ) 交替符号梯形的数量与 l - 1 类的列严格移位平面分区的数量相同，顶行中最多 n 个部分，从而证明了独立推测的结果贝伦德和艾格纳。 [2]