Conjugating full cycles by adjacent transpositions: diameter and sorting time
Ron M. Adin, Eli Bagno, Yuval Roichman
TL;DR
The paper analyzes the problem of transforming an $n$-cycle to the canonical cycle via conjugation by adjacent transpositions, recasting it as a Schreier-graph problem on left cosets of $\mathbb{Z}_n$ in $S_n$ through the minv statistic. It derives quadratic bounds for both the Coxeter cyclic sorting time $\operatorname{Sort}_n$ and the diameter of the associated graph $\Gamma_n$, by developing weighted inversions ($\operatorname{winv}$) and its cyclic variant ($\operatorname{cwinv}$) to bound expectations and then translating these into worst-case distances. The main contributions include precise upper and lower bounds: $\frac{8-\pi}{16}\cdot n^2+O(n) \le \operatorname{Sort}_n \le \frac{1}{3}\cdot n^2+O(n)$ and $\frac{8-\pi}{16}\cdot n^2+O(n) \le \operatorname{Diameter}(\Gamma_n) \le \frac{3}{8}\cdot n^2+O(n)$, along with a detailed diameter bound of $\operatorname{Diameter}(\Gamma_n) \le \frac{3n^2-4n+1}{8}$. The authors also connect these questions to the structure of the Schreier graph $X(S_n/{\mathbb{Z}}_n,\Phi_n)$ and discuss open problems, including the exact asymptotics and cyclic extensions, with implications for sorting in permutation groups and related order structures.
Abstract
We establish upper and lower bounds on the maximal number of steps needed to transform a cyclic permutation to the canonical cyclic permutation using conjugation by adjacent transpositions, and on the diameter of the underlying Schreier graph.