对于$q$ -ary字符串，具有到根的转置距离的复制
Duplication with transposition distance to the root for $q$-ary strings
摘要：
研究了$q$元字母表上长度为$n$的字符串与其根之间的换位距离重复问题。换句话说，我们研究了$x = (abcd) \to y = (abcbd)$形式的重复操作的数量，其中$x$和$y$是字符串，$a$，$b$，$c$和$d$是它们的子串，需要从没有重复的字符串集合开始得到长度为$n$的$q$ -ary字符串。对于精确复制，我们证明了长度至多为$n$的字符串与其根之间的最大距离具有渐近阶$n/\log n$。对于近似重复，其中一个$β$ -分数的符号可能会被错误地重复，我们表明，最大的距离有一个急剧的过渡，从$n/\log n$的顺序$\log n$在$β=(q-1)/q$。这个问题的动机来自基因组学，其中这种重复代表一种特殊类型的突变，并且给定生物序列与其根之间的距离是产生序列所需的最小数量的转座突变。
英文摘要：
We study the duplication with transposition distance between strings of length $n$ over a $q$-ary alphabet and their roots. In other words, we investigate the number of duplication operations of the form $x = (abcd) \to y = (abcbd)$, where $x$ and $y$ are strings and $a$, $b$, $c$ and $d$ are their substrings, needed to get a $q$-ary string of length $n$ starting from the set of strings without duplications. For exact duplication, we prove that the maximal distance between a string of length at most $n$ and its root has the asymptotic order $n/\log n$. For approximate duplication, where a $β$-fraction of symbols may be duplicated incorrectly, we show that the maximal distance has a sharp transition from the order $n/\log n$ to $\log n$ at $β=(q-1)/q$. The motivation for this problem comes from genomics, where such duplications represent a special kind of mutation and the distance between a given biological sequence and its root is the smallest number of transposition mutations required to generate the sequence.
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