The Transposition Distance Problem (TDP) is a classical problem in genome rearrangements which seeks to determine the minimum number of transpositions needed to transform a linear chromosome into another represented by the permutations $\pi$ and $\sigma$. This paper focuses on the equivalent problem of Sorting By Transpositions (SBT), where $\sigma$ is the identity permutation $\iota$. Specifically, we investigate properties of palisades, a family of permutations that are ``hard'' to sort, as they require numerous transpositions above the celebrated lower bound devised by Bafna and Pevzner. By determining the transposition distance of palisades, we were able to provide the exact transposition diameter for $3$-permutations (TD3), a special subset of the Symmetric Group $S_n$, essential for the study of approximate solutions for SBT using the simplification technique. The exact value for TD3 has remained unknown since Elias and Hartman showed an upper bound for it. Another consequence of determining the transposition distance of palisades is that, using as lower bound the one by Bafna and Pevzner, it is impossible to guarantee approximation ratios lower than $1.375$ when approximating SBT. This finding has significant implications for the study of SBT, as this problem has been subject of intense research efforts for the past 25 years.

翻译：转座距离问题（TDP）是基因组重排中的一个经典问题，旨在确定将一条线性染色体转化为另一条由排列 $\pi$ 和 $\sigma$ 表示的染色体所需的最小转座次数。本文聚焦于等价问题——转座排序（SBT），其中 $\sigma$ 为单位排列 $\iota$。具体而言，我们研究了“栅栏”排列的性质，这是一类难以排序的排列，因为它们所需的转座次数远超 Bafna 和 Pevzner 提出的著名下界。通过确定栅栏排列的转座距离，我们得以给出 $3$-排列（TD3）的精确转座直径，这是对称群 $S_n$ 的一个特殊子集，对于利用简化技术研究 SBT 的近似解至关重要。自 Elias 和 Hartman 给出其上界以来，TD3 的精确值一直未知。确定栅栏排列转座距离的另一结果是，以 Bafna 和 Pevzner 的下界作为基准，在近似 SBT 时无法保证低于 $1.375$ 的近似比。这一发现对 SBT 的研究具有重要意义，因为该问题在过去 25 年间一直是学界深入研究的焦点。