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 $\pi\in {\rm{Sym}_n}$ is a word (product of elements) of $S$. A positive integer $d(\pi)$ is associated with each $\pi\in{\rm{Sym}_n}$ taking the length of the shortest such word, and the $S$-diameter $d(S)$ is the maximum value of $d(\pi)$ with $\pi$ ranging over ${\rm{Sym_n}}$. The distance $d(\pi,\nu)$ of two permutations $\pi,\nu$ defined by $d(\nu^{-1}\circ\pi)$ 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 $d(\pi)$ the block transposition distance of $\pi$. A strong motivation for the study of this special case comes from investigations of large-scale mutations of genome, where determining $d(\pi)$ is known as sorting the permutation $\pi$ by block transpositions. In the papers on this subject, toric equivalence classes often play a crucial role since $d(\pi)=d(\nu)$ when $\pi$ and $\nu$ 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 A proof of this result can be found in the (unpublished) Hausen's Ph.D Dissertation thesis; see \cite{Ha}.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.