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 $\sigma$, respectively. This paper focuses on the equivalent problem of Sorting By Transpositions (SBT), where $\sigma$ is the identity permutation $\iota$. Specifically, we investigate 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）是基因组重排中的经典问题，旨在确定将线性染色体转化为由排列π和σ表示的另一染色体所需的最少转座次数。本文聚焦于等价问题——通过转座排序（SBT），其中σ为单位排列ι。具体而言，我们研究了栅栏排列（palisades）这一族“难以”排序的排列，因其所需转座次数远超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的特有子集，对于利用简化技术研究SBT近似解至关重要。自Elias和Hartman给出TD3的上界以来，其精确值一直未知。确定栅栏排列转座距离的另一结果是，若以Bafna和Pevzner的下界为基准，近似SBT时无法保证低于1.375的近似比。这一发现对SBT研究具有深远意义，因为该问题在过去25年间一直是深入研究的热点。