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| Format: | Preprint |
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2025
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| Online-Zugang: | https://arxiv.org/abs/2510.18529 |
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| _version_ | 1866915591488536576 |
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| author | Bastide, Paul Bishnoi, Anurag Groenland, Carla Gijswijt, Dion Joshi, Rohinee |
| author_facet | Bastide, Paul Bishnoi, Anurag Groenland, Carla Gijswijt, Dion Joshi, Rohinee |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_18529 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Circular sorting, strong complete mappings and wreath product constructions Bastide, Paul Bishnoi, Anurag Groenland, Carla Gijswijt, Dion Joshi, Rohinee Combinatorics 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. |
| title | Circular sorting, strong complete mappings and wreath product constructions |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2510.18529 |