A DNA string is a Watson-Crick (WK) palindrome when the complement of its reverse is equal to itself. The Watson-Crick mapping $\theta$ is an involution that is also an antimorphism. $\theta$-conjugates of a word is a generalisation of conjugates of a word that incorporates the notion of WK-involution $\theta$. In this paper, we study the distribution of palindromes and Watson-Crick palindromes, also known as $\theta$-palindromes among both the set of conjugates and $\theta$-conjugates of a word $w$. We also consider some general properties of the set $C_{\theta}(w)$, i.e., the set of $\theta$-conjugates of a word $w$, and characterize words $w$ such that $|C_{\theta}(w)|=|w|+1$, i.e., with the maximum number of elements in $C_{\theta}(w)$. We also find the structure of words that have at least one (WK)-palindrome in $C_{\theta}(w)$.

中文翻译：

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θ 共轭中的 Theta 回文

当其反向的补数等于自身时，DNA 串是 Watson-Crick (WK) 回文。Watson-Crick 映射 $\theta$ 是一个对合，也是一个反态。一个词的 $\theta$-共轭是一个词的共轭的概括，它结合了 WK-involution $\theta$ 的概念。在本文中，我们研究了回文和 Watson-Crick 回文的分布，也称为 $\theta$-回文在单词 $w$ 的共轭集和 $\theta$-共轭中的分布。我们还考虑了集合 $C_{\theta}(w)$ 的一些一般性质，即词 $w$ 的 $\theta$-共轭集合，并刻画词 $w$ 使得 $|C_{ \theta}(w)|=|w|+1$，即$C_{\theta}(w)$ 中元素的最大数目。我们还在 $C_{\theta}(w)$ 中找到了至少有一个 (WK)-回文的单词的结构。