A word is said to be \emph{bordered} if it contains a non-empty proper prefix that is also a suffix. We can naturally extend this definition to pairs of non-empty words. A pair of words $(u,v)$ is said to be \emph{mutually bordered} if there exists a word that is a non-empty proper prefix of $u$ and suffix of $v$, and there exists a word that is a non-empty proper suffix of $u$ and prefix of $v$. In other words, $(u,v)$ is mutually bordered if $u$ overlaps $v$ and $v$ overlaps $u$. We give a recurrence for the number of mutually bordered pairs of words. Furthermore, we show that, asymptotically, there are $c\cdot k^{2n}$ mutually bordered words of length-$n$ over a $k$-letter alphabet, where $c$ is a constant. Finally, we show that the expected shortest overlap between pairs of words is bounded above by a constant.
翻译：如果一个单词含有非空白的正确前缀, 且该前缀也包含一个后缀。 我们自然可以将这个定义扩展至非空白的单词。 一对单词 $( u, v), 美元据说是 $( u, v) 和 $( $) 重叠 $( u) 。 如果一个单词是非空白的正确前缀, 美元和 v 美元, 则该单词被称为 $( u, v) 。 如果存在一个非空白的正确前缀, 美元和 $( v), 则该单词被称为 $( u, v) 和 $( $) 重叠 $( $) 。 如果一个单词有非空白的正确前缀, 美元和 $( v) $( $)? 则该单词是非空白的正确前缀, 则该单词是非空白的正确后缀, $( $) $( n) 和 $( $( $c) $) $( $) 美元) 等前缀是非空白的正确后缀。 。 。 。, 换换,, 美元是相互连接的单词是相同的。 最后, 我们将显示每字中最短的预期的重叠。
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