Upper bound on the block transposition diameter of the symmetric group

Abstract

Given a generator set $S$ of the symmetric group ${\rm{Sym}}_n$, every permutation $π\in {\rm{Sym}_n}$ is a word (product of elements) of $S$. A positive integer $d(π)$ is associated with each $π\in{\rm{Sym}_n}$ taking the length of the shortest such word, and the $S$-diameter $d(S)$ is the maximum value of $d(π)$ with $π$ ranging over ${\rm{Sym_n}}$. The distance $d(π,ν)$ of two permutations $π,ν$ defined by $d(ν^{-1}\circπ)$ satisfies the axioms of a metric space. In this paper we consider the case where $S$ consists of all block transpositions of ${\rm{Sym_n}}$ and call $d(π)$ the block transposition distance of $π$. A strong motivation for the study of this special case comes from investigations of large-scale mutations of genome, where determining $d(π)$ is known as sorting the permutation $π$ by block transpositions. In the papers on this subject, toric equivalence classes often play a crucial role since $d(π)=d(ν)$ when $π$ and $ν$ are torically equivalent. A proof of this result can be found in the (unpublished) Hausen's Ph.D Dissertation thesis; see \cite{Ha}. Our main contribution is to obtain a bijective map on ${\rm{Sym}_n}$ from the toric equivalence that leaves the distances invariant. Using the properties of this map, we give an alternative proof of Hausen's result which actually fills a gap in the proof of the upper bound on $d(S)$ due to Eriksson and his coworkers; see \cite{EE}. We also revisit the proof of the key lemma \cite[Lemma 5,1]{EE}, giving more details and filling some gaps.