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：换位距离问题（TDP）是基因组重排中的经典问题，旨在确定将一条线性染色体转化为另一条染色体所需的最小换位次数，这两条染色体分别由排列π和σ表示。本文聚焦于等价问题——互换排序（SBT），其中σ为单位排列ι。具体而言，我们研究了护栏排列（palisades）这一族“难以”排序的排列，因为它们所需的换位次数远超Bafna与Pevzner提出的著名下界。通过确定护栏排列的换位距离，我们得以给出3-排列（TD3）的精确换位直径。3-排列是对称群S_n的特殊子集，对于利用简化技术研究SBT的近似解至关重要。自Elias与Hartman给出其上界以来，TD3的精确值一直未知。确定护栏排列换位距离的另一项意义在于：以Bafna与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的研究具有重要启示，因为该问题在过去25年间一直是深入研究的焦点。