Upper bound on the block transposition diameter of the symmetric group

TL;DR

This paper establishes an upper bound on the block transposition diameter of the symmetric group, introduces a bijective map preserving distances, and provides an alternative, detailed proof filling gaps in prior work.

Contribution

It introduces a bijective map on permutations that preserves block transposition distances and offers a new proof of the upper bound on the diameter, addressing gaps in earlier proofs.

Findings

The bijective map preserves toric equivalence classes and distances.

An alternative proof of the upper bound on the block transposition diameter is provided.

The paper clarifies and fills gaps in previous proofs of key lemmas.

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 A proof of this result can be found in the (unpublished) Hausen's Ph.D Dissertation thesis; see \cite{Ha}.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.