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Reference: Ian M. Wanless, (2001). Answers to questions by Dénes on Latin power sets. European Journal of Combinatorics, 22 (7), 1009-1020.Citable link to this page:

 

Answers to questions by Dénes on Latin power sets

Abstract: The ith power, Li, of a Latin square L is that matrix obtained by replacing each row permutation in L by its ith power. A Latin power set of cardinality m ≥ 2 is a set of Latin squares {A, A2, A3, … , Am}. We prove some basic properties of Latin power sets and use them to resolve questions asked by Dénes and his various collaborators. Dénes has used Latin power sets in an attempt to settle a conjecture by Hall and Paige on complete mappings in groups. Dénes suggested three generalisations of the Hall–Paige conjecture. We refute all three with counterexamples. Elsewhere, Dénes et al. unsuccessfully tried to construct three mutually orthogonal Latin squares of order 10 based on a Latin power set. We confirm as a result of an exhaustive computer search that there is no Latin power set of the kind sought. However we do find a set of four mutually orthogonal 9 × 10 Latin rectangles.We also show the non-existence of a 2-fold perfect (10, 9, 1)-Mendelsohn design which was conjectured to exist by Dénes. Finally, we prove a conjecture originally due to Dénes and Keedwell and show that two others of Dénes and Owens are false.

Publication status:PublishedPeer Review status:Peer reviewedVersion:Publisher's versionNotes:Copyright 2001 Academic Presss, an imprint of Elsevier B.V. All rights reserved. Re-use of this article is permitted in accordance with the Terms and Conditions set out at http://www.elsevier.com/open-access/userlicense/1.0/

Bibliographic Details

Publisher: Elsevier B.V.

Publisher Website: http://www.elsevier.com/

Host: European Journal of Combinatoricssee more from them

Publication Website: http://www.sciencedirect.com/science/journal/01956698

Issue Date: 2001-October

Copyright Date: 2001

pages:1009-1020Identifiers

Doi: https://doi.org/10.1006/eujc.2001.0518

Issn: 0195-6698

Urn: uuid:78a3c1d3-2927-4ee6-ad16-a26b66380af3 Item Description

Type: Article: post-print;

Language: en

Version: Publisher's versionKeywords: Magic squares Set theory Dénes, J. (József)Subjects: Mathematics Combinatorics Tiny URL: ora:8505

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Author: Ian M. Wanless - institutionUniversity of Oxford facultyMathematical,Physical and Life Sciences Division oxfordCollegeChrist Chur

Source: https://ora.ox.ac.uk/objects/uuid:78a3c1d3-2927-4ee6-ad16-a26b66380af3



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