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Mathematics of Computation

Published by the American Mathematical Society, the Mathematics of Computation (MCOM) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.98.

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Enumeration of racks and quandles up to isomorphism
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by Petr Vojtěchovský and Seung Yeop Yang HTML | PDF
Math. Comp. 88 (2019), 2523-2540 Request permission

Abstract:

Racks and quandles are prominent set-theoretical solutions of the Yang-Baxter equation. We enumerate racks and quandles of orders $n\le 13$ up to isomorphism, improving upon the previously known results for $n\le 8$ and $n\le 9$, respectively. The enumeration is based on the classification of subgroups of small symmetric groups up to conjugation, on a representation of racks and quandles in symmetric groups due to Joyce and Blackburn, and on a number of theoretical and computational observations concerning the representation. We explicitly find representatives of isomorphism types of racks of order $\le 11$ and quandles of order $\le 12$. For the remaining orders we merely count the isomorphism types, relying in part on the enumeration of $2$-reductive racks and $2$-reductive quandles due to Jedlička, Pilitowska, Stanovský, and Zamojska-Dzienio.
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Additional Information
  • Petr Vojtěchovský
  • Affiliation: Department of Mathematics, University of Denver, 2390 S York St, Denver, Colorado, 80208
  • MR Author ID: 650320
  • Email: petr@math.du.edu
  • Seung Yeop Yang
  • Affiliation: Department of Mathematics, Kyungpook National University, Daegu, 41566, Republic of Korea
  • MR Author ID: 1111587
  • Email: seungyeop.yang@knu.ac.kr
  • Received by editor(s): May 22, 2018
  • Received by editor(s) in revised form: September 25, 2018
  • Published electronically: January 7, 2019
  • Additional Notes: The first author was supported by a 2015 PROF grant of the University of Denver.
  • © Copyright 2019 American Mathematical Society
  • Journal: Math. Comp. 88 (2019), 2523-2540
  • MSC (2010): Primary 16T25, 20N05, 57M27
  • DOI: https://doi.org/10.1090/mcom/3409
  • MathSciNet review: 3957904