Counting racks of order n
A rack on [n] can be thought of as a set of maps (f x )x∈ [n] , where each f x is a permutation of [n] such that f (x) f y =f −1 y f x f y for all x and y. In 2013, Blackburn showed that the number of isomorphism classes of racks on [n][n] is at least 2 (1/4−o(1)) n 2 and at most 2 (c+o(1)) n 2 , w...
| Auteurs principaux: | , |
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| Format: | Journal article |
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Electronic Journal of Combinatorics
2017
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| _version_ | 1826302335888916480 |
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| author | Ashford, M Riordan, O |
| author_facet | Ashford, M Riordan, O |
| author_sort | Ashford, M |
| collection | OXFORD |
| description | A rack on [n] can be thought of as a set of maps (f x )x∈ [n] , where each f x is a permutation of [n] such that f (x) f y =f −1 y f x f y for all x and y. In 2013, Blackburn showed that the number of isomorphism classes of racks on [n][n] is at least 2 (1/4−o(1)) n 2 and at most 2 (c+o(1)) n 2 , where c≈1.557; in this paper we improve the upper bound to 2 (1/4+o(1)) n 2 , matching the lower bound. The proof involves considering racks as loopless, edge-coloured directed multigraphs on [n], where we have an edge of colour y between x and z if and only if (x)f y =z, and applying various combinatorial tools. |
| first_indexed | 2024-03-07T05:46:00Z |
| format | Journal article |
| id | oxford-uuid:e73d15ff-c49a-463c-ad6e-2283c3f9adcf |
| institution | University of Oxford |
| last_indexed | 2024-03-07T05:46:00Z |
| publishDate | 2017 |
| publisher | Electronic Journal of Combinatorics |
| record_format | dspace |
| spelling | oxford-uuid:e73d15ff-c49a-463c-ad6e-2283c3f9adcf2022-03-27T10:37:07ZCounting racks of order nJournal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:e73d15ff-c49a-463c-ad6e-2283c3f9adcfSymplectic Elements at OxfordElectronic Journal of Combinatorics2017Ashford, MRiordan, OA rack on [n] can be thought of as a set of maps (f x )x∈ [n] , where each f x is a permutation of [n] such that f (x) f y =f −1 y f x f y for all x and y. In 2013, Blackburn showed that the number of isomorphism classes of racks on [n][n] is at least 2 (1/4−o(1)) n 2 and at most 2 (c+o(1)) n 2 , where c≈1.557; in this paper we improve the upper bound to 2 (1/4+o(1)) n 2 , matching the lower bound. The proof involves considering racks as loopless, edge-coloured directed multigraphs on [n], where we have an edge of colour y between x and z if and only if (x)f y =z, and applying various combinatorial tools. |
| spellingShingle | Ashford, M Riordan, O Counting racks of order n |
| title | Counting racks of order n |
| title_full | Counting racks of order n |
| title_fullStr | Counting racks of order n |
| title_full_unstemmed | Counting racks of order n |
| title_short | Counting racks of order n |
| title_sort | counting racks of order n |
| work_keys_str_mv | AT ashfordm countingracksofordern AT riordano countingracksofordern |