For each $\alpha \in \{0,1,-1 \}$, we count diagonally and antidiagonally symmetric alternating sign matrices (DASASMs) of fixed odd order with a maximal number of $\alpha$'s along the diagonal and the antidiagonal, as well as DASASMs of fixed odd order with a minimal number of $0$'s along the diagonal and the antidiagonal. In these enumerations, we encounter product formulas that have previously appeared in plane partition or alternating sign matrix counting, namely for the number of all alternating sign matrices, the number of cyclically symmetric plane partitions in a given box, and the number of vertically and horizontally symmetric ASMs. We also prove several refinements. For instance, in the case of DASASMs with a maximal number of $-1$'s along the diagonal and the antidiagonal, these considerations lead naturally to the definition of alternating sign triangles. These are new objects that are equinumerous with ASMs, and we are able to prove a two parameter refinement of this fact, involving the number of $-1$'s and the inversion number on the ASM side. To prove our results, we extend techniques to deal with triangular six-vertex configurations that have recently successfully been applied to settle Robbins' conjecture on the number of all DASASMs of odd order. Importantly, we use a general solution of the reflection equation to prove the symmetry of the partition function in the spectral parameters. In all of our cases, we derive determinant or Pfaffian formulas for the partition functions, which we then specialize in order to obtain the product formulas for the various classes of extreme odd DASASMs under consideration.

中文翻译：

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奇数阶极对角和对角对称交替符号矩阵

对于每个 $\alpha \in \{0,1,-1 \}$，我们计算固定奇数阶的对角和对角对称交替符号矩阵（DASASM），其中最大数量的 $\alpha$ 沿着对角线和对角线，以及沿对角线和对角线具有最小数量的 $0$ 的固定奇数阶的 DASASM。在这些枚举中，我们遇到了以前出现在平面分区或交替符号矩阵计数中的乘积公式，即对于所有交替符号矩阵的数量，给定框中循环对称平面分区的数量，以及垂直和水平方向的数量对称 ASM。我们还证明了一些改进。例如，在 DASASM 的情况下，沿对角线和对角线的最大数量为 $-1$，这些考虑自然导致交替符号三角形的定义。这些是与 ASM 相同的新对象，我们能够证明这一事实的两个参数细化，涉及 $-1$ 的数量和 ASM 端的反转数。为了证明我们的结果，我们扩展了处理三角形六顶点配置的技术，这些技术最近已成功应用于解决罗宾斯关于所有奇数阶 DASASM 数量的猜想。重要的是，我们使用反射方程的通解来证明光谱参数中分配函数的对称性。在我们所有的情况下，我们推导出分区函数的行列式或 Pfaffian 公式，