We present and prove closed form expressions for some families of binomial determinants with signed Kronecker deltas that are located along an arbitrary diagonal in the corresponding matrix. They count cyclically symmetric rhombus tilings of hexagonal regions with triangular holes. We extend a previous systematic study of these families, where the locations of the Kronecker deltas depended on an additional parameter, to families with negative Kronecker deltas. By adapting Zeilberger's holonomic ansatz to make it work for our problems, we can take full advantage of computer algebra tools for symbolic summation. This, together with the combinatorial interpretation, allows us to realize some new determinantal relationships. From there, we are able to resolve all remaining open conjectures related to these determinants, including one from 2005 due to Lascoux and Krattenthaler.

中文翻译：

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关于平铺问题的二项式行列式产生完整的Ansatz

我们提出并证明带符号的Kronecker三角洲的二项式行列式的某些族的闭合形式表达式，这些三角族位于相应矩阵中的任意对角线上。他们计算带有三角形孔的六边形区域的周期性对称菱形平铺。我们将先前对这些家庭的系统研究扩展到负Kronecker三角洲的家庭，其中Kronecker三角洲的位置取决于其他参数。通过调整Zeilberger的完整ansatz使其适用于我们的问题，我们可以充分利用计算机代数工具进行符号求和。这与组合解释一起使我们能够实现一些新的确定性关系。从那里，我们能够解决所有与这些决定因素有关的未解决猜想，