A two-dimensional string is simply a two-dimensional array. We continue the study of the combinatorial properties of repetitions in such strings over the binary alphabet, namely the number of distinct tandems, distinct quartics, and runs. First, we construct an infinite family of $n\times n$ 2D strings with $\Omega(n^{3})$ distinct tandems. Second, we construct an infinite family of $n\times n$ 2D strings with $\Omega(n^{2}\log n)$ distinct quartics. Third, we construct an infinite family of $n\times n$ 2D strings with $\Omega(n^{2}\log n)$ runs. This resolves an open question of Charalampopoulos, Radoszewski, Rytter, Wale\'n, and Zuba [ESA 2020], who asked if the number of distinct quartics and runs in an $n\times n$ 2D string is $\mathcal{O}(n^{2})$.

中文翻译：

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二维字符串中重复次数的下限

二维字符串只是一个二维数组。我们继续研究二进制字母表上此类字符串中重复的组合特性，即不同串联、不同四次和游程的数量。首先，我们构造了一个具有 $\Omega(n^{3})$ 不同串联的 $n\times n$ 二维字符串的无限族。其次，我们用 $\Omega(n^{2}\log n)$ 不同的四次方构造了一个 $n\times n$ 二维字符串的无限族。第三，我们用 $\Omega(n^{2}\log n)$ 运行构建了一个 $n\times n$ 二维字符串的无限族。这解决了 Charalampopoulos、Radoszewski、Rytter、Wale\'n 和 Zuba [ESA 2020] 的一个悬而未决的问题，他们询问在 $n\times n$ 二维字符串中运行的不同四次的数量是否为 $\mathcal{O }(n^{2})$。