Circular sorting, strong complete mappings and wreath product constructions
Paul Bastide, Anurag Bishnoi, Carla Groenland, Dion Gijswijt, Rohinee Joshi

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 labelled points on a circle by transpositions. Imagine that the vertices of a cycle of length are labelled by the elements . 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 in clockwise order? We provide evidence for their conjecture that at most transpositions are needed to sort a circular permutation when is not prime. We prove this conjecture when or and when restricting to permutations given by a polynomial over . We also provide various algebraic constructions of circular permutations that take many transpositions to sort, most notably providing one that matches our upper bound…
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Taxonomy
TopicsGenome Rearrangement Algorithms · Advanced Combinatorial Mathematics · Limits and Structures in Graph Theory
