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.

我们提出并证明了一些二项式行列式的闭式表达式，这些行列式具有带符号的 Kronecker deltas，它们位于相应矩阵中的任意对角线上。他们计算具有三角形孔的六边形区域的循环对称菱形平铺。我们将先前对这些家族的系统研究（其中克罗内克三角洲的位置取决于一个附加参数）扩展到具有负克罗内克三角洲的家庭。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认识到一些新的决定性关系。从那里，我们能够解决与这些决定因素相关的所有剩余的开放猜想，