A091325 -id:A091325 - cp's OEIS frontend

A090741 Maximum number of transversals in a Latin square of order n.

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

Offset: 1

Richard Bean, Feb 03 2004

a(10) >= 5504 from Parker.

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

			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.

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.

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

Cf. A006717, A091325.

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

a(9) = 2241 from Brendan McKay and Ian Wanless, May 23 2004

A091326 Total number of inequivalent even binary linear codes of length n and any dimension k = 0, 1, ..., n-1.

Original entry on oeis.org

1, 2, 3, 6, 9, 18, 31, 64, 123

Offset: 1

N. J. A. Sloane, Mar 01 2004

"Even" means that every word has even weight. Equivalently, the all-ones vector is in the dual code.

G. Nebe, E. M. Rains and N. J. A. Sloane, Self-Dual Codes and Invariant Theory, Springer, Berlin, 2006.
Index entries for sequences related to binary linear codes

Row sums of A091325.

A091327 Triangle T(n,k) read by rows giving number of inequivalent even binary linear [n,k] codes containing no zero columns (n >= 2, 1 <= k <= n-1).

Original entry on oeis.org

1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 2, 3, 2, 1, 0, 2, 4, 4, 2, 1, 1, 3, 8, 10, 7, 3, 1, 0, 3, 9, 18, 16, 9, 3, 1

Offset: 2

N. J. A. Sloane, Mar 01 2004

"Even" means that every word has even weight. Equivalently, the all-ones vector is in the dual code.

Difference between successive rows of A091325.

			Triangle begins
1
0 1
1 1 1
0 1 1 1
1 2 3 2 1
0 2 4 4 2 1

G. Nebe, E. M. Rains and N. J. A. Sloane, Self-Dual Codes and Invariant Theory, Springer, Berlin, 2006.
Index entries for sequences related to binary linear codes

Cf. A091325. Row sums give A091328.

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A090741 Maximum number of transversals in a Latin square of order n.

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A091326 Total number of inequivalent even binary linear codes of length n and any dimension k = 0, 1, ..., n-1.

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Author

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Comments

Links

Crossrefs

A091327 Triangle T(n,k) read by rows giving number of inequivalent even binary linear [n,k] codes containing no zero columns (n >= 2, 1 <= k <= n-1).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs