论文标题
交替的符号矩阵和完全对称的平面分区
Alternating sign matrices and totally symmetric plane partitions
论文作者
论文摘要
我们介绍了一个新的家族$ \ MATHCAL {a} _ {n，k} $的schur阳性对称函数，该功能定义为完全对称平面分区的总和。在第一部分中，我们表明，对于$ k = 1 $，该家族等于涉及$ n+3 $的对象变量的多元生成功能，这些函数扩展了交替的符号矩阵（ASMS），这些功能最近由作者引入。这在ASM和一类平面分区之间建立了新的联系，从而补充了ASM与完全对称的自相互平面分区以及下降的平面分区的事实。该证明是基于新的反对称器到确定的公式，我们还为其提供了徒证明。在第二部分中，我们将$ \ MATHCAL {a} _ {n，k} $的三个专业化与某些众所周知的列类别的严格移位平面分区的加权枚举相关联，这些枚举将延伸到下降平面分区。
We introduce a new family $\mathcal{A}_{n,k}$ of Schur positive symmetric functions, which are defined as sums over totally symmetric plane partitions. In the first part, we show that, for $k=1$, this family is equal to a multivariate generating function involving $n+3$ variables of objects that extend alternating sign matrices (ASMs), which have recently been introduced by the authors. This establishes a new connection between ASMs and a class of plane partitions, thereby complementing the fact that ASMs are equinumerous with totally symmetric self-complementary plane partitions as well as with descending plane partitions. The proof is based on a new antisymmetrizer-to-determinant formula for which we also provide a bijective proof. In the second part, we relate three specialisation of $\mathcal{A}_{n,k}$ to a weighted enumeration of certain well-known classes of column strict shifted plane partitions that generalise descending plane partitions.
