A266166 -id:A266166 - cp's OEIS frontend

A129732 Number of main classes of Graeco-Latin squares of order n. That is, species of pairs of orthogonal Latin squares.

1, 1, 1, 0, 7, 2165, 91846374

Offset: 3

Brendan McKay, May 13 2007

			The unique example for n=3 has two Latin squares [[0, 2, 1], [2, 1, 0], [1, 0, 2]] and [[0, 2, 1], [1, 0, 2], [2, 1, 0]].

J. Egan and I. M. Wanless, Enumeration of MOLS of small order, Mathematics of Computation 85, 2016, 799-824.
B. D. McKay, Latin squares.
I. Wanless, Data on mutually orthogonal Latin squares (MOLS).

Cf. A003090, A266166, A266167, A266168, A266169, A266170, A266171, A266172, A266173.

a(9) by Egan and Wanless added by Janne I. Kokkala, Sep 08 2015

A266167 Number of isotopism classes of unordered pairs of orthogonal Latin squares.

Original entry on oeis.org

1, 0, 1, 1, 2, 0, 20, 23362, 1101734942

Offset: 1

Ian Wanless, Dec 22 2015

The following operations produce things that are counted as equivalent: simultaneous permutation of the rows or columns of both squares; permutation of the symbols within either square; interchanging the squares.

This sequence is also the number of trisotopism classes of *ordered* pairs of orthogonal Latin squares, where the following operations produce things that are counted as equivalent: simultaneous permutation of the rows or columns of both squares; permutation of the symbols within either square; transposing both squares.

J. Egan and I. M. Wanless, Enumeration of MOLS of small order, Mathematics of Computation 85, 2016, 799-824.
Ian Wanless, Data on MOLS

Cf. A072377, A129732, A266166, A266168, A266169.

A266168 Number of isotopism classes of ordered pairs of orthogonal Latin squares.

Original entry on oeis.org

1, 0, 1, 1, 3, 0, 34, 45927, 2203310919

Offset: 1

Ian Wanless, Dec 22 2015

The following operations produce things that are counted as equivalent: simultaneous permutation of the rows or columns of both squares; permutation of the symbols within either square.

J. Egan and I. M. Wanless, Enumeration of MOLS of small order, Mathematics of Computation 85, 2016, 799-824.
Ian Wanless, Data on MOLS

Cf. A129732, A266166, A266167.

A072377 Number of pairs of orthogonal Latin squares of order n.

Original entry on oeis.org

1, 0, 36, 3456, 3110400, 0, 3131834388480000, 32162058365970677760000, 19083454070548282639185346560000

Offset: 1

Eric W. Weisstein, Jul 19 2002

This sequence counts unordered pairs. The number of ordered pairs of MOLS is exactly double this sequence.

J. Egan and I. M. Wanless, Enumeration of MOLS of small order, Mathematics of Computation 85, 2016, 799-824.
Ian Wanless, Data on MOLS
Eric Weisstein's World of Mathematics, Latin Square

Equals n!n!(n-1)!/2 times A266166 (except for the degenerate case n=1). Cf. A000315, A002860, A266167, A266168, A266169, A266170, A266171, A266172, A266173.

More terms (and corrected the degenerate first term) from Ian Wanless, Dec 22 2015

A266169 Number of trisotopism classes of unordered pairs of orthogonal Latin squares.

Original entry on oeis.org

1, 0, 1, 1, 2, 0, 14, 11887, 550907773

Offset: 1

Ian Wanless, Dec 22 2015

The following operations produce things that are counted as equivalent: simultaneous permutation of the rows or columns of both squares; permutation of the symbols within either square; interchanging the squares; transposing both squares.

J. Egan and I. M. Wanless, Enumeration of MOLS of small order, Mathematics of Computation 85, 2016, 799-824.
Ian Wanless, Data on MOLS

Cf. A266166, A266167, A266168.

A266171 Number of reduced maximal pairs of orthogonal Latin squares.

Original entry on oeis.org

0, 0, 1, 0, 0, 0, 341880, 7832534400, 7188529229970480

Offset: 1

Ian Wanless, Dec 23 2015

A pair of MOLS is reduced if their first rows are in natural order and so is the first column of the first square. The pair is maximal if it cannot be extended to a triple of MOLS.

J. Egan and I. M. Wanless, Enumeration of MOLS of small order, Mathematics of Computation 85, 2016, 799-824.
Ian Wanless, Data on MOLS

Cf. A072377, A266166, A266170.

A287651 Number of reduced pairs of orthogonal diagonal Latin squares of order n.

Original entry on oeis.org

1, 0, 0, 2, 4, 0, 320, 1322496, 339930624

Offset: 1

Eduard I. Vatutin, May 29 2017

A pair of orthogonal diagonal Latin squares is reduced if their first rows are in natural order.

E. I. Vatutin, Discussion about properties of diagonal Latin squares at forum.boinc.ru (in Russian)
E. I. Vatutin, Discussion about properties of diagonal Latin squares at forum.boinc.ru, continuation (in Russian)
Eduard I. Vatutin, Stepan E. Kochemazov, Oleq S. Zaikin, Maxim O. Manzuk, Natalia N. Nikitina, Vitaly S. Titov, Central symmetry properties for diagonal Latin squares, Problems of Information Technology (2019) No. 2, 3-8.
E. I. Vatutin, S. E. Kochemazov, O. S. Zaikin, M. O. Manzuk, V. S. Titov, Combinatorial characteristics estimating for pairs of orthogonal diagonal Latin squares, Multicore processors, parallel programming, FPGA, signal processing systems (2017), pp. 104-111 (in Russian).
Eduard I. Vatutin, Natalia N. Nikitina, Maxim O. Manzuk, First results of an experiment on studying the properties of DLS of order 9 in the volunteer distributed computing projects Gerasim@Home and RakeSearch (in Russian).
Eduard I. Vatutin, Natalia N. Nikitina, Maxim O. Manzuk, Additional calculated results of an experiment on studying the properties of DLS of order 9 in the volunteer distributed computing projects Gerasim@Home and RakeSearch (in Russian).
Index entries for sequences related to Latin squares and rectangles

Cf. A266166, A339926.

a(n) = A339926(n) / (n!)^2. - Eduard I. Vatutin, Dec 24 2020

a(8) added by Eduard I. Vatutin, Jan 02 2018

a(9) added by Eduard I. Vatutin, Dec 22 2020

cp's OEIS Frontend

A129732 Number of main classes of Graeco-Latin squares of order n. That is, species of pairs of orthogonal Latin squares.

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A266167 Number of isotopism classes of unordered pairs of orthogonal Latin squares.

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A266168 Number of isotopism classes of ordered pairs of orthogonal Latin squares.

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A072377 Number of pairs of orthogonal Latin squares of order n.

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A266169 Number of trisotopism classes of unordered pairs of orthogonal Latin squares.

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A266171 Number of reduced maximal pairs of orthogonal Latin squares.

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A287651 Number of reduced pairs of orthogonal diagonal Latin squares of order n.

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