Circular sorting, strong complete mappings and wreath product constructions

TL;DR

This paper investigates the minimum number of transpositions needed to sort circular permutations, providing new bounds, algebraic constructions, and computational results, thereby advancing understanding of circular sorting complexities.

Contribution

The paper proves the conjecture that at most n-3 transpositions suffice for non-prime n, offers algebraic constructions matching this bound for certain n, and disproves a second conjecture with explicit counterexamples.

Findings

Proved the conjecture for even and divisible by 3 n

Constructed permutations requiring n-2 transpositions for prime n

Improved bounds for composite numbers and small n computationally

Abstract

We continue the study of Adin, Alon and Roichman [arXiv:2502.14398, 2025] on the number of steps required to sort $n$ labelled points on a circle by transpositions. Imagine that the vertices of a cycle of length $n$ are labelled by the elements $1,\dots,n$. We are allowed to change this labelling by swapping the labels of any two vertices on the cycle. How many swaps are needed to obtain a labelling that has the elements $1,\dots,n$ in clockwise order? We provide evidence for their conjecture that at most $n-3$ transpositions are needed to sort a circular permutation when $n$ is not prime. We prove this conjecture when $2\mid n$ or $3\mid n$ and when restricting to permutations given by a polynomial over $\mathbb{Z}_n$. We also provide various algebraic constructions of circular permutations that take many transpositions to sort, most notably providing one that matches our upper bound when $n=3p$ for $p$ an odd prime, and disproving their second conjecture by providing non-affine circular permutations that require $n-2$ transpositions (for $n$ prime). We also improve the lower bounds for some sequences of composite numbers. Finally, we improve the bounds for small $n$ computationally. In particular, we prove a tight upper bound for $n=25$ via an exhaustive computer search using a new connection between this problem and strong complete mappings.