A338522 -id:A338522 - cp's OEIS frontend

A338562 Number of cyclic diagonal Latin squares of order 2n+1.

1, 0, 240, 20160, 0, 319334400, 62270208000, 0, 4979623993344000, 1946321606541312000, 0, 517040334777699532800000, 155112100433309859840000000, 0, 229885811837232250818134016000000, 230239482316981838896315760640000000, 0, 82665183731089159437333210700185600000000

Offset: 0

Eduard I. Vatutin, Nov 02 2020

A cyclic Latin square is a Latin square in which row i is obtained by cyclically shifting row i-1 by d places.
Equivalently, a Latin square is cyclic if and only if each row is a cyclic permutation of the first row and each column is a cyclic permutation of the first column.
Every cyclic diagonal Latin square is a cyclic Latin square, so a(n) <= A338522(2*n+1).
Cyclic diagonal Latin squares exist only for odd orders not divisible by 3. - Andrew Howroyd, May 26 2021

			For n=3 there are 6 cyclic Latin squares of order 7 with the first row in ascending order, only 4 of them are diagonal:
  0 1 2 3 4 5 6   0 1 2 3 4 5 6   0 1 2 3 4 5 6   0 1 2 3 4 5 6
  2 3 4 5 6 0 1   3 4 5 6 0 1 2   4 5 6 0 1 2 3   5 6 0 1 2 3 4
  4 5 6 0 1 2 3   6 0 1 2 3 4 5   1 2 3 4 5 6 0   3 4 5 6 0 1 2
  6 0 1 2 3 4 5   2 3 4 5 6 0 1   5 6 0 1 2 3 4   1 2 3 4 5 6 0
  1 2 3 4 5 6 0   5 6 0 1 2 3 4   2 3 4 5 6 0 1   6 0 1 2 3 4 5
  3 4 5 6 0 1 2   1 2 3 4 5 6 0   6 0 1 2 3 4 5   4 5 6 0 1 2 3
  5 6 0 1 2 3 4   4 5 6 0 1 2 3   3 4 5 6 0 1 2   2 3 4 5 6 0 1
and 4*7! = 20160 cyclic diagonal Latin squares.

Eduard I. Vatutin, Enumerating cyclic Latin squares and Euler totient function calculating using them, High-performance computing systems and technologies, 2020, Vol. 4, No. 2, pp. 40-48. (in Russian)
Eduard I. Vatutin, Numerical formula between number of cyclic diagonal Latin squares and number of toroidal n-queens problem solutions getting by knight movement (in Russian).
E. I. Vatutin, Special types of diagonal Latin squares, Cloud and distributed computing systems in electronic control conference, within the National supercomputing forum (NSCF - 2022). Pereslavl-Zalessky, 2023. pp. 9-18. (in Russian)
Index entries for sequences related to Latin squares and rectangles.

Cf. A123565 (ordered first row), A338522, A341585 (main classes), A342306, A370672.

a(n)={my(m=2*n+1); m!*if(gcd(m, 6)==1, sum(k=1, m, gcd(k^3-k, m)==1))} \\ Andrew Howroyd, Apr 30 2021

a(n) = A123565(2*n+1) * (2*n+1)!.

a(n) = A370672(n) * (2n)!. - Eduard I. Vatutin, Mar 13 2024

More terms from Andrew Howroyd, Apr 30 2021

Zero terms for even orders removed by Andrew Howroyd, May 26 2021

A372922 Number of diagonal Latin squares of order 2n+1 that are isomorphic to cyclic Latin squares by row and column permutations.

Original entry on oeis.org

1, 0, 480, 161280, 2229534720, 45984153600000, 3271798279766016000

Offset: 0

Eduard I. Vatutin, May 16 2024

The Latin squares considered here are diagonal Latin squares that are isomorphic to cyclic Latin squares. They are can be obtained from cyclic Latin squares (see A338522) by diagonalization (getting a corresponding pair of transversals and placing them on the diagonals, see article). These Latin squares have some interesting properties, for example, there are a large number of diagonal transversals.

			The cyclic Latin square of order 7
.
  0 1 2 3 4 5 6
  1 2 3 4 5 6 0
  2 3 4 5 6 0 1
  3 4 5 6 0 1 2
  4 5 6 0 1 2 3
  5 6 0 1 2 3 4
  6 0 1 2 3 4 5
.
has a pair of symmetrically placed transversals T1 = (0, 2, 4, 6, 1, 3, 5) and T2 = (0, 5, 3, 1, 6, 4, 2), after permutting rown and columns transversal T1 placed to the main diagonal with getting single diagonal Latin square
.
  2 5 0 3 4 6 1
  0 3 5 1 2 4 6
  1 4 6 2 3 5 0
  6 2 4 0 1 3 5
  3 6 1 4 5 0 2
  4 0 2 5 6 1 3
  5 1 3 6 0 2 4
.
then after permuting rows and columns transversal T2 placed to the second diagonal with getting diagonal Latin square
.
  2 5 0 3 6 1 4
  0 3 5 1 4 6 2
  1 4 6 2 5 0 3
  6 2 4 0 3 5 1
  4 0 2 5 1 3 6
  5 1 3 6 2 4 0
  3 6 1 4 0 2 5
.
that can be canonized to the following diagonal Latin square:
.
  0 1 2 3 4 5 6
  2 3 1 5 6 4 0
  5 6 4 0 1 2 3
  4 0 6 2 3 1 5
  6 2 0 1 5 3 4
  1 5 3 4 0 6 2
  3 4 5 6 2 0 1
.
Cyclic Latin square of order 11
.
  0 1 2 3 4 5 6 7 8 9 10
  1 2 3 4 5 6 7 8 9 10 0
  2 3 4 5 6 7 8 9 10 0 1
  3 4 5 6 7 8 9 10 0 1 2
  4 5 6 7 8 9 10 0 1 2 3
  5 6 7 8 9 10 0 1 2 3 4
  6 7 8 9 10 0 1 2 3 4 5
  7 8 9 10 0 1 2 3 4 5 6
  8 9 10 0 1 2 3 4 5 6 7
  9 10 0 1 2 3 4 5 6 7 8
  10 0 1 2 3 4 5 6 7 8 9
.
can be diagonalized to set of diagonal Latin squares:
.
  0 1 2 3 4 5 6 7 8 9 10   0 1 2 3 4 5 6 7 8 9 10   0 1 2 3 4 5 6 7 8 9 10
  1 2 3 4 5 10 8 9 0 6 7   1 2 3 4 6 7 8 9 10 0 5   1 2 3 4 5 10 9 0 7 8 6
  3 4 5 10 7 9 1 8 2 0 6   8 10 5 7 9 3 0 4 1 6 2   3 4 5 10 6 9 7 2 1 0 8
  4 5 10 7 9 6 2 0 3 1 8   4 6 8 10 5 1 7 2 9 3 0   10 6 9 8 7 0 2 5 4 3 1
  10 7 9 6 8 0 4 2 5 3 1   9 0 1 2 3 10 4 5 6 7 8   9 8 7 0 1 2 4 6 10 5 3
  7 9 6 8 0 1 5 3 10 4 2   7 9 0 1 2 8 3 10 4 5 6   5 10 6 9 8 7 1 4 3 2 0
  8 0 1 2 3 4 9 10 6 7 5   6 8 10 5 7 2 9 3 0 4 1   7 0 1 2 3 4 10 8 9 6 5
  2 3 4 5 10 7 0 6 1 8 9   10 5 7 9 0 4 1 6 2 8 3   4 5 10 6 9 8 0 3 2 1 7
  5 10 7 9 6 8 3 1 4 2 0   3 4 6 8 10 0 5 1 7 2 9   8 7 0 1 2 3 5 9 6 10 4
  6 8 0 1 2 3 7 5 9 10 4   2 3 4 6 8 9 10 0 5 1 7   6 9 8 7 0 1 3 10 5 4 2
  9 6 8 0 1 2 10 4 7 5 3   5 7 9 0 1 6 2 8 3 10 4   2 3 4 5 10 6 8 1 0 7 9 ...
.
(totally 81 main classes of diagonal Latin squares).

Eduard I. Vatutin, Special types of diagonal Latin squares, Cloud and distributed computing systems in electronic control conference, within the National supercomputing forum (NSCF - 2022). Pereslavl-Zalessky, 2023. pp. 9-18. (in Russian)
Eduard I. Vatutin, About the different types of cyclic diagonal Latin squares (in Russian).
E. Vatutin, A. Belyshev, N. Nikitina, M. Manzuk, A. Albertian, I. Kurochkin, A. Kripachev, and A. Pykhtin, Diagonalization and Canonization of Latin Squares, Lecture Notes in Computer Science, Vol. 14389, Springer, Cham., 2023. pp. 48-61.
Index entries for sequences related to Latin squares and rectangles.

Cf. A338522, A338562, A342990, A372923, A375475.

a(n) = A372923(n) * (2n+1)!. - Eduard I. Vatutin, Sep 08 2024

A372923 Number of diagonalized cyclic diagonal Latin squares of order 2n+1 with the first row in order.

Original entry on oeis.org

1, 0, 4, 32, 6144, 1152000, 45984153600000

Offset: 0

Eduard I. Vatutin, May 16 2024

See Comments in A372922.

Eduard I. Vatutin, Special types of diagonal Latin squares, Cloud and distributed computing systems in electronic control conference, within the National supercomputing forum (NSCF - 2022). Pereslavl-Zalessky, 2023. pp. 9-18. (in Russian)
Eduard I. Vatutin, About the different types of cyclic diagonal Latin squares (in Russian).
E. Vatutin, A. Belyshev, N. Nikitina, M. Manzuk, A. Albertian, I. Kurochkin, A. Kripachev, and A. Pykhtin, Diagonalization and Canonization of Latin Squares, Lecture Notes in Computer Science, Vol. 14389, Springer, Cham., 2023. pp. 48-61.
Index entries for sequences related to Latin squares and rectangles.

Cf. A071607, A123565, A338522, A372922, A375475.

a(n) = A372922(n) / (2n+1)!. - Eduard I. Vatutin, Sep 08 2024

A375475 Number of main classes of diagonalized cyclic diagonal Latin squares of order 2n+1.

Original entry on oeis.org

1, 0, 1, 1, 7, 81, 2933

Offset: 0

Eduard I. Vatutin, Aug 17 2024

Diagonalized cyclic diagonal Latin squares are diagonal Latin squares that are isomorphic to cyclic Latin squares. They are can be obtained from cyclic Latin squares (see A338522) by diagonalization (getting a corresponding pair of transversals and placing them on the diagonals, see article). Diagonalized cyclic diagonal Latin squares have some interesting properties, for example, there are a large number of diagonal transversals for diagonal Latin squares of odd orders.

Eduard I. Vatutin, Special types of diagonal Latin squares, Cloud and distributed computing systems in electronic control conference, within the National supercomputing forum (NSCF - 2022). Pereslavl-Zalessky, 2023. pp. 9-18. (in Russian)
Eduard I. Vatutin, About the different types of cyclic diagonal Latin squares (in Russian).
E. Vatutin, A. Belyshev, N. Nikitina, M. Manzuk, A. Albertian, I. Kurochkin, A. Kripachev, and A. Pykhtin, Diagonalization and Canonization of Latin Squares, Lecture Notes in Computer Science, Vol. 14389, Springer, Cham., 2023. pp. 48-61.
Proving lists.
Index entries for sequences related to Latin squares and rectangles.

Cf. A338522, A338562, A372922, A372923.

A383368 Number of intercalates in pine Latin squares of order 2n.

Original entry on oeis.org

1, 12, 27, 80, 125, 252, 343, 576, 729, 1100, 1331, 1872, 2197, 2940, 3375, 4352, 4913, 6156, 6859, 8400, 9261, 11132, 12167, 14400, 15625

Offset: 1

Eduard I. Vatutin, Apr 24 2025

Pine Latin square is a none canonical composite Latin square of order N=2*K formed from specially arranged cyclic Latin squares of order K.
By construction, pine Latin square is determined one-to-one by the cyclic square used, so number of pine Latin squares of order N is equal to number of cyclic Latin squares of order N/2.
All pine Latin squares are horizontally symmetric column-inverse Latin squares.
All pine Latin squares for selected order N are isomorphic one to another as Latin squares, so they have same properties (number of transversals, intercalates, etc.).
Pine Latin squares have interesting properties, for example, maximum known number of intercalates for some orders N (at least N in {2, 4, 6, 10, 18}).
Pine Latin squares do not exist for odd orders due to they are horizontally symmetric.
Pine Latin squares of order N=2n exists for all even orders due to existing of corresponding cyclic Latin squares of order n. According to this, maximum number of intercalates in a Latin square A092237(N) >= (2k)^2 * (2k + 1) for N=4k and A092237(N) >= (2k+1)^3 for N=4k+2. Therefore, asimptotically maximum number of intercalates in Latin squares of even orders N greater or equal than o(k1*N^3), where k1 = 1/8.

			For order N=8 pine Latin square
  0 1 2 3 4 5 6 7
  1 2 3 0 7 4 5 6
  2 3 0 1 6 7 4 5
  3 0 1 2 5 6 7 4
  4 5 6 7 0 1 2 3
  5 6 7 4 3 0 1 2
  6 7 4 5 2 3 0 1
  7 4 5 6 1 2 3 0
have 80 intercalates.
.
For order N=10 pine Latin square
  0 1 2 3 4 5 6 7 8 9
  1 2 3 4 0 9 5 6 7 8
  2 3 4 0 1 8 9 5 6 7
  3 4 0 1 2 7 8 9 5 6
  4 0 1 2 3 6 7 8 9 5
  5 6 7 8 9 0 1 2 3 4
  6 7 8 9 5 4 0 1 2 3
  7 8 9 5 6 3 4 0 1 2
  8 9 5 6 7 2 3 4 0 1
  9 5 6 7 8 1 2 3 4 0
have 125 intercalates.
.
For order N=12 pine Latin square
  0 1 2 3 4 5 6 7 8 9 10 11
  1 2 3 4 5 0 11 6 7 8 9 10
  2 3 4 5 0 1 10 11 6 7 8 9
  3 4 5 0 1 2 9 10 11 6 7 8
  4 5 0 1 2 3 8 9 10 11 6 7
  5 0 1 2 3 4 7 8 9 10 11 6
  6 7 8 9 10 11 0 1 2 3 4 5
  7 8 9 10 11 6 5 0 1 2 3 4
  8 9 10 11 6 7 4 5 0 1 2 3
  9 10 11 6 7 8 3 4 5 0 1 2
  10 11 6 7 8 9 2 3 4 5 0 1
  11 6 7 8 9 10 1 2 3 4 5 0
have 252 intercalates.

R. Bean, Critical sets in Latin squares and associated structures, Ph.D. Thesis, The University of Queensland, 2001.
Eduard I. Vatutin, About the properties of pine Latin squares (in Russian).
Eduard I. Vatutin, Proving list (examples).
Index entries for sequences related to Latin squares and rectangles.

Cf. A002860, A016755, A089207, A092237, A099721, A338522, A383570.

Hypothesis: For all known pine Latin squares of orders N=4k+2 number of intercalates a(n) = a(N/2)= a(2k+1) = (N/2)^3 = (2k+1)^3 = A016755((n-1)/2) (verified for N<29).

Hypothesis: For all known pine Latin squares of orders N=4k number of intercalates a(n) = a(N/2) = a(2k) = (N/2)^2 + (N/2)^3 = 4*k^2 + 8*k^3 = (2k)^2 * (2k+1) = 2*A089207(n/2) = 4*A099721(n/2) (verified for N<29).

A383570 Number of transversals in pine Latin squares of order 4n.

Original entry on oeis.org

8, 384, 76032, 62881792

Offset: 1

Eduard I. Vatutin, Apr 30 2025

A pine Latin square is a not necessarily canonical composite Latin square of order N=2*K formed from specially arranged cyclic Latin squares of order K.
By construction, pine Latin square is determined one-to-one by the cyclic square used, so number of pine Latin squares of order N is equal to number of cyclic Latin squares of order N/2.
All pine Latin squares are horizontally symmetric column-inverse Latin squares.
All pine Latin squares for selected order N are isomorphic one to another as Latin squares, so they have same properties (number of transversals, intercalates, etc.).
Pine Latin squares have interesting properties, for example, maximum known number of intercalates (see A383368 and A092237) for some orders N (at least N in {2, 4, 6, 10, 18}).
Pine Latin squares do not exist for odd orders because they must be horizontally symmetric.
Hypothesis: number of transversals in pine Latin squares of all orders N=4k+2 is zero (verified for orders N<=18).

			For order N=8 pine Latin square
  0 1 2 3 4 5 6 7
  1 2 3 0 7 4 5 6
  2 3 0 1 6 7 4 5
  3 0 1 2 5 6 7 4
  4 5 6 7 0 1 2 3
  5 6 7 4 3 0 1 2
  6 7 4 5 2 3 0 1
  7 4 5 6 1 2 3 0
has 384 transversals.
.
For order N=10 pine Latin square
  0 1 2 3 4 5 6 7 8 9
  1 2 3 4 0 9 5 6 7 8
  2 3 4 0 1 8 9 5 6 7
  3 4 0 1 2 7 8 9 5 6
  4 0 1 2 3 6 7 8 9 5
  5 6 7 8 9 0 1 2 3 4
  6 7 8 9 5 4 0 1 2 3
  7 8 9 5 6 3 4 0 1 2
  8 9 5 6 7 2 3 4 0 1
  9 5 6 7 8 1 2 3 4 0
has no transversals.
.
For order N=12 pine Latin square
  0 1 2 3 4 5 6 7 8 9 10 11
  1 2 3 4 5 0 11 6 7 8 9 10
  2 3 4 5 0 1 10 11 6 7 8 9
  3 4 5 0 1 2 9 10 11 6 7 8
  4 5 0 1 2 3 8 9 10 11 6 7
  5 0 1 2 3 4 7 8 9 10 11 6
  6 7 8 9 10 11 0 1 2 3 4 5
  7 8 9 10 11 6 5 0 1 2 3 4
  8 9 10 11 6 7 4 5 0 1 2 3
  9 10 11 6 7 8 3 4 5 0 1 2
  10 11 6 7 8 9 2 3 4 5 0 1
  11 6 7 8 9 10 1 2 3 4 5 0
has 76032 transversals.

Richard Bean, Critical sets in Latin squares and associated structures, Ph.D. Thesis, The University of Queensland, 2001.
Eduard I. Vatutin, About the properties of pine Latin squares (in Russian).
Eduard I. Vatutin, Proving list (examples).
Index entries for sequences related to Latin squares and rectangles.

Cf. A002860, A091323, A090741, A338522, A383368.

cp's OEIS Frontend

A338562 Number of cyclic diagonal Latin squares of order 2n+1.

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A372922 Number of diagonal Latin squares of order 2n+1 that are isomorphic to cyclic Latin squares by row and column permutations.

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A372923 Number of diagonalized cyclic diagonal Latin squares of order 2n+1 with the first row in order.

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A375475 Number of main classes of diagonalized cyclic diagonal Latin squares of order 2n+1.

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A383368 Number of intercalates in pine Latin squares of order 2n.

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A383570 Number of transversals in pine Latin squares of order 4n.

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