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Maximum number of transversals in a Latin square of order n.
7

%I #48 Feb 28 2024 13:21:29

%S 1,0,3,8,15,32,133,384,2241

%N Maximum number of transversals in a Latin square of order n.

%C a(10) >= 5504 from Parker.

%C a(n) >= the number of transversals in a cyclic Latin square of the same order which for odd n is given by A006717((n-1)/2). - _Eduard I. Vatutin_, Nov 04 2020

%D 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.

%D E. T. Parker, Computer investigations of orthogonal Latin squares of order 10, Proc. Sympos. Appl. Math., volume 15 (1963), 73-81.

%H D. Bedford, <a href="http://dx.doi.org/10.1016/S0195-6698(13)80096-0">Transversals in the Cayley tables of the non-cyclic groups of order 8</a>, European Journal of Combinatorics, volume 12 (1991), 455-458.

%H N. J. Cavenagh and I. M. Wanless, <a href="http://dx.doi.org/10.1016/j.dam.2009.09.006">On the number of transversals in Cayley tables of cyclic groups</a>, Disc. Appl. Math. 158 (2010), 136-146.

%H B. D. McKay, J. C. McLeod and I. M. Wanless, <a href="http://dx.doi.org/10.1007/s10623-006-0012-8">The number of transversals in a Latin square</a>, Des. Codes Cryptogr., 40, (2006) 269-284.

%H V. N. Potapov, <a href="https://arxiv.org/abs/1506.01577">On the number of transversals in Latin squares</a>, arxiv:1506.01577 [math.CO], 2015.

%H Eduard Vatutin, Alexey Belyshev, Natalia Nikitina, Maxim Manzuk, Alexander Albertian, Ilya Kurochkin, Alexander Kripachev, and Alexey Pykhtin, <a href="https://doi.org/10.1007/978-3-031-49435-2_4">Diagonalization and Canonization of Latin Squares</a>, Supercomputing, Russian Supercomputing Days (RuSCDays 2023) Rev. Selected Papers Part II, LCNS Vol. 14389, Springer, Cham, 48-61.

%H Ian M. Wanless, <a href="https://doi.org/10.37236/1629">A Generalization of Transversals for Latin Squares</a>, Electronic Journal of Combinatorics, volume 9, number 1 (2002), R12.

%H <a href="/index/La#Latin">Index entries for sequences related to Latin squares and rectangles</a>

%F a(n) is asymptotically in between 3.2^n and 0.62^n n!. [McKay, McLeod, Wanless], [Cavenagh, Wanless]. - _Ian Wanless_, Jul 30 2010

%e 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).

%e a(9) = 2241 from Z_3 x Z_3.

%Y Cf. A006717, A091325.

%K hard,more,nonn

%O 1,3

%A _Richard Bean_, Feb 03 2004

%E a(9) = 2241 from _Brendan McKay_ and _Ian Wanless_, May 23 2004