This paper addresses the question of how intercalates occur in the two known infinite families of triple arrays, the Paley triple arrays constructed in 2005 by Preece et al., and the Triple arrays from difference sets in 2017 by Nilson and Cameron. The main reason for doing this is that the number of such embedded Latin squares is often used when checking whether two arrays are isotopic or not. We determine sharp bounds for the number and density of intercalates for the main subclasses of these families respectively. We also prove the existence of an infinite family of triple arrays in which every two occurrences of an entry lie in an intercalate.

中文翻译：

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插入双列和三列阵列

本文解决了在两个已知的无限三重阵列家族中如何发生插层的问题，即 Preece 等人于 2005 年构建的 Paley 三重阵列，以及 Nilson 和 Cameron 在 2017 年构建的来自不同集合的三重阵列。这样做的主要原因是，在检查两个阵列是否为同位素时，经常使用这种嵌入的拉丁方格的数量。我们分别为这些家族的主要子类确定了插层的数量和密度的明确界限。我们还证明了一个无限的三重数组家族的存在，其中每两次出现的条目都位于一个插层中。