A090741 Maximum number of transversals in a Latin square of order n.
1, 0, 3, 8, 15, 32, 133, 384, 2241
Offset: 1
Examples
a(1), a(3), a(5), a(7) are from the group tables for Z_1, Z_3, Z_5 and Z_7 (see sequence A006717); a(4) and a(8) are from Z_2 x Z_2 and the non-cyclic groups of order 8 (see Bedford). a(9) = 2241 from Z_3 x Z_3.
References
- J. W. Brown, F. Cherry, L. Most, M. Most, E. T. Parker, W. D. Wallis, Completion of the spectrum of orthogonal diagonal Latin squares, Lecture notes in pure and applied mathematics. 1992. Vol. 139. pp. 43-49.
- E. T. Parker, Computer investigations of orthogonal Latin squares of order 10, Proc. Sympos. Appl. Math., volume 15 (1963), 73-81.
Links
- D. Bedford, Transversals in the Cayley tables of the non-cyclic groups of order 8, European Journal of Combinatorics, volume 12 (1991), 455-458.
- N. J. Cavenagh and I. M. Wanless, On the number of transversals in Cayley tables of cyclic groups, Disc. Appl. Math. 158 (2010), 136-146.
- B. D. McKay, J. C. McLeod and I. M. Wanless, The number of transversals in a Latin square, Des. Codes Cryptogr., 40, (2006) 269-284.
- V. N. Potapov, On the number of transversals in Latin squares, arxiv:1506.01577 [math.CO], 2015.
- Eduard Vatutin, Alexey Belyshev, Natalia Nikitina, Maxim Manzuk, Alexander Albertian, Ilya Kurochkin, Alexander Kripachev, and Alexey Pykhtin, Diagonalization and Canonization of Latin Squares, Supercomputing, Russian Supercomputing Days (RuSCDays 2023) Rev. Selected Papers Part II, LCNS Vol. 14389, Springer, Cham, 48-61.
- Ian M. Wanless, A Generalization of Transversals for Latin Squares, Electronic Journal of Combinatorics, volume 9, number 1 (2002), R12.
- Index entries for sequences related to Latin squares and rectangles
Formula
a(n) is asymptotically in between 3.2^n and 0.62^n n!. [McKay, McLeod, Wanless], [Cavenagh, Wanless]. - Ian Wanless, Jul 30 2010
Extensions
a(9) = 2241 from Brendan McKay and Ian Wanless, May 23 2004
Comments