Advertisement

Designs, Codes and Cryptography

, Volume 40, Issue 3, pp 269–284 | Cite as

The number of transversals in a Latin square

  • Brendan D. McKay
  • Jeanette C. McLeod
  • Ian M. Wanless
Article

Abstract

A Latin Square of order n is an n ×  n array of n symbols, in which each symbol occurs exactly once in each row and column. A transversal is a set of n entries, one selected from each row and each column of a Latin Square of order n such that no two entries contain the same symbol. Define T(n) to be the maximum number of transversals over all Latin squares of order n. We show that \(b^n \leq T(n) \leq c^n\sqrt{n}\,n!\) for n ≥ 5, where b ≈ 1.719 and c ≈ 0.614. A corollary of this result is an upper bound on the number of placements of n non-attacking queens on an n ×  n toroidal chess board. Some divisibility properties of the number of transversals in Latin squares based on finite groups are established. We also provide data from a computer enumeration of transversals in all Latin Squares of order at most 9, all groups of order at most 23 and all possible turn-squares of order 14.

Keywords

Transversal Latin square n-queens Turn-square Cayley table 

AMS Classification

05B15 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Akbari, S, Alireza, A 2004Transversals and multicolored matchingsJ Combin Des12325332zbMATHMathSciNetCrossRefGoogle Scholar
  2. 2.
    Balasubramanian, K 1990On transversals in Latin SquaresLinear Algebra Appl131125129zbMATHMathSciNetCrossRefGoogle Scholar
  3. 3.
    Bedford, D, Whitaker, RM 1999Enumeration of transversals in the Cayley tables of the non-cyclic groups of order 8Discrete Math197/1987781MathSciNetGoogle Scholar
  4. 4.
    Brown, JW, Parker, ET 1993More on order 10 turn-squaresArs Combin35125127zbMATHMathSciNetGoogle Scholar
  5. 5.
    Cameron, PJ, Wanless, IM 2005Covering radius for sets of permutationsDiscrete Math29391109zbMATHMathSciNetCrossRefGoogle Scholar
  6. 6.
    Cooper, C 2000A lower bound for the number of good permutationsData Recording, Storage and Processing (Nat. Acad. Sci. Ukraine)2131525Google Scholar
  7. 7.
    Cooper, C, Gilchrist, R, Kovalenko, I, Novakovic, D 2000Deriving the number of good permutations, with application to cryptographyCybern Syst Anal51016zbMATHMathSciNetGoogle Scholar
  8. 8.
    Cooper, C, Kovalenko, I 1996The upper bound for the number of complete mappingsTheory Probab Math Statist537783MathSciNetGoogle Scholar
  9. 9.
    Dénes, J, Keedwell, AD 1974Latin Squares and their applicationsAkadémiai KiadóBudapestzbMATHGoogle Scholar
  10. 10.
    Dénes, J, Keedwell, AD 1991Latin Squares: new developments in the theory and applicationsNorth-HollandAmsterdamAnnator of Discrete Mathametics, vol. 46Google Scholar
  11. 11.
    Derienko II (1988) On a conjecture of Brualdi (in Russian). Mat Issled. No. 102, Issled. Oper. i Kvazigrupp 119:53–65Google Scholar
  12. 12.
    Fu, H, Lin, S 2002The length of a partial transversal in a Latin SquareJ Combin Math Combin Comput435764zbMATHMathSciNetGoogle Scholar
  13. 13.
    Heinrich K (1976) Latin Squares composed of four disjoint subsquares. Combinatorial mathematics V. Proceedings of Fifth Australian Conference Royal Melbourne Institute of Technology, Melbourne, pp 118–127Google Scholar
  14. 14.
    Hsiang J, Shieh Y, Chen Y (2002) The cyclic complete mappings counting problems, PaPS: Problems and Problem Sets for ATP Workshop in conjunction with CADE-18 and FLoC 2002, Copenhagen, DenmarkGoogle Scholar
  15. 15.
    Killgrove, R, Roberts, C, Sternfeld, R, Tamez, R, Derby, R, Kiel, D 1996Latin Squares and other configurationsCongr Numer117161174zbMATHMathSciNetGoogle Scholar
  16. 16.
    Kovalenko, I 1996On an upper bound for the number of complete mappingsCybern Syst Anal326568zbMATHMathSciNetCrossRefGoogle Scholar
  17. 17.
    Maenhaut, BM, Wanless, IM 2004Atomic Latin Squares of order elevenJ Combin Des121234zbMATHMathSciNetCrossRefGoogle Scholar
  18. 18.
    McKay BD, Meynert A, Myrvold W (to appear) Small Latin squares, quasigroups and loops. J Combin Des http://dx.doi.org/10.1002/jed.2005Google Scholar
  19. 19.
    McKay, BD, Wanless, IM 1999Most Latin Squares have many subsquaresJ Combin Theory Ser A86323347zbMATHMathSciNetCrossRefGoogle Scholar
  20. 20.
    Rivin, I, Vardi, I, Zimmerman, P 1994The n-queens problemAm Math Monthly101629639zbMATHCrossRefGoogle Scholar
  21. 21.
    Ryser, HJ 1967Neuere probleme der KombinatorikVortrage über Kombinatorik Oberwolfach24–296991Google Scholar
  22. 22.
    Shieh YP (2001) Partition Strategies for #P-complete problem with applications to enumerative combinatorics. PhD thesis, National Taiwan UniversityGoogle Scholar
  23. 23.
    Shieh, YP, Hsiang, J, Hsu, DF 2000On the enumeration of abelian k-complete mappingsCongr Numer1446788zbMATHMathSciNetGoogle Scholar
  24. 24.
    Shor, PW 1982A lower bound for the length of a partial transversal in a Latin SquareJ Combin Theory Ser A3318zbMATHMathSciNetCrossRefGoogle Scholar
  25. 25.
    Thomas, AD, Wood, GV 1980Group tablesShiva PublishingNantwichShiva Mathematics Series 2zbMATHGoogle Scholar
  26. 26.
    Vardi, I 1991Computational recreations in mathematicsAddison-WesleyRedwood City, CAGoogle Scholar
  27. 27.
    Vaughan-Lee, M, Wanless, IM 2003Latin Squares and the Hall-Paige conjectureBull Lond Math Soc3515zbMATHMathSciNetGoogle Scholar
  28. 28.
    Wanless, IM 2002A generalisation of transversals for Latin SquaresElectron J Combin9R12zbMATHMathSciNetGoogle Scholar

Copyright information

© Springer Science + Business Media, LLC 2006

Authors and Affiliations

  • Brendan D. McKay
    • 1
  • Jeanette C. McLeod
    • 1
  • Ian M. Wanless
    • 1
    • 2
  1. 1.Computer Science DepartmentAustralian National UniversityCanberraAustralia
  2. 2.School of Mathematical SciencesMonash UniversityVicAustralia

Personalised recommendations