A123565 -id:A123565 - cp's OEIS frontend

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

A343866 Number of inequivalent cyclic diagonal Latin squares of order 2n+1 up to rotations, reflections and permutation of symbols.

Original entry on oeis.org

1, 0, 1, 1, 0, 2, 3, 0, 4, 4, 0, 5, 3, 0, 7, 7, 0, 2, 9, 0, 10, 10, 0, 11, 7, 0, 13, 4, 0, 14, 15, 0, 6, 16, 0, 17, 18, 0, 8, 19, 0, 20, 8, 0, 22, 10, 0, 8, 24, 0, 25, 25, 0, 26, 27, 0, 28, 10, 0, 14, 22, 0, 13, 31, 0, 32, 16, 0, 34, 34, 0, 20, 14, 0, 37, 37, 0, 14, 39, 0, 20

Offset: 0

Andrew Howroyd, May 02 2021

nonn

Also the number of main classes of diagonal Latin squares of order 2n+1 that contain a cyclic Latin square. Compare A341585.

			a(12) = 3 since there are A123565(25) = 10 cyclic diagonal Latin squares whose first row is in ascending order. Each of these is uniquely defined by the step between rows and form 5 pairs by horizontal or vertical reflection (negating the step between rows). Up to exchanging rows with columns there are 3 distinct classes, so a(12) = 3.

Cf. A123565, A287764, A338562, A341585.

iscanon(n,k,g) = k <= vecmin(g*k%n) && k <= vecmin(g*lift(1/Mod(k,n))%n)
a(n)={if(n==0, 1, my(m=2*n+1); sum(k=1, m-1, gcd(m,k)==1 && gcd(m,k-1)==1 && gcd(m,k+1)==1 && iscanon(m, k, [1,-1])))}

a((p-1)/2) = A341585((p-1)/2) for odd prime p.

A366331 Number of main classes of diagonal Latin squares of order 2n+1 that contain a horizontally semicyclic Latin square.

Original entry on oeis.org

1, 0, 1, 1, 0, 2, 20, 0, 272, 1208, 0, 127334, 1958084, 0

Offset: 0

Eduard I. Vatutin, Oct 07 2023

A horizontally semicyclic diagonal Latin square is a square where each row r(i) is a cyclic shift of the first row r(0) by some value d(i) (see example).

			Example of horizontally semicyclic diagonal Latin square of order 13:
   0  1  2  3  4  5  6  7  8  9 10 11 12
   2  3  4  5  6  7  8  9 10 11 12  0  1  (d=2)
   4  5  6  7  8  9 10 11 12  0  1  2  3  (d=4)
   9 10 11 12  0  1  2  3  4  5  6  7  8  (d=9)
   7  8  9 10 11 12  0  1  2  3  4  5  6  (d=7)
  12  0  1  2  3  4  5  6  7  8  9 10 11  (d=12)
   3  4  5  6  7  8  9 10 11 12  0  1  2  (d=3)
  11 12  0  1  2  3  4  5  6  7  8  9 10  (d=11)
   6  7  8  9 10 11 12  0  1  2  3  4  5  (d=6)
   1  2  3  4  5  6  7  8  9 10 11 12  0  (d=1)
   5  6  7  8  9 10 11 12  0  1  2  3  4  (d=5)
  10 11 12  0  1  2  3  4  5  6  7  8  9  (d=10)
   8  9 10 11 12  0  1  2  3  4  5  6  7  (d=8)

Eduard I. Vatutin, About the horizontally and vertically semicyclic diagonal Latin squares enumeration (in Russian).
Eduard I. Vatutin, About the spectra of numerical characteristics of different types of cyclic diagonal Latin squares (in Russian).
Eduard I. Vatutin, About the number of main classes of semicyclic diagonal Latin squares of order 17 (in Russian).
Eduard I. Vatutin, About the number of main classes of semicyclic diagonal Latin squares of order 19 (in Russian).
Eduard I. Vatutin, Lists of canonical forms of semicyclic diagonal Latin squares of orders 5-19.
Index entries for sequences related to Latin squares and rectangles.

Cf. A071607, A123565, A287764, A338562, A342990, A343866.

a(11)-a(13) from Andrew Howroyd, Nov 02 2023

A370672 Number of ways of arranging 2n+1 nonattacking queens on a 2n+1 X 2n+1 toroidal board using knight moves.

Original entry on oeis.org

1, 0, 10, 28, 0, 88, 130, 0, 238, 304, 0, 460, 250, 0, 754, 868, 0, 280, 1258, 0, 1558, 1720, 0, 2068, 1372, 0, 2650, 880, 0, 3304, 3538, 0, 1300, 4288, 0, 4828, 5110, 0, 2464, 6004, 0, 6640, 2380, 0, 7654, 3640, 0

Offset: 0

Eduard I. Vatutin, Feb 25 2024

nonn

All solutions of this type can be found using a knight moving with some displacements dx and dy starting from some cell with coordinates (x,y): (x,y) -> (x+dx,y+dy) -> (x+2*dx,y+2*dy) -> ... -> (x,y) (all operations modulo n). For n <= 11 all solutions of n nonattacking queens on n X n a toroidal board problem are solutions of this type, for n >= 13 some solutions are not of this type (see A051906 for examples).

			For n=2*2+1=5 there are 10 solutions:
.
+-----------+ +-----------+ +-----------+ +-----------+ +-----------+
| Q . . . . | | Q . . . . | | . Q . . . | | . Q . . . | | . . Q . . |
| . . Q . . | | . . . Q . | | . . . Q . | | . . . . Q | | Q . . . . |
| . . . . Q | | . Q . . . | | Q . . . . | | . . Q . . | | . . . Q . |
| . Q . . . | | . . . . Q | | . . Q . . | | Q . . . . | | . Q . . . |
| . . . Q . | | . . Q . . | | . . . . Q | | . . . Q . | | . . . . Q |
+-----------+ +-----------+ +-----------+ +-----------+ +-----------+
.
+-----------+ +-----------+ +-----------+ +-----------+ +-----------+
| . . Q . . | | . . . Q . | | . . . Q . | | . . . . Q | | . . . . Q |
| . . . . Q | | Q . . . . | | . Q . . . | | . Q . . . | | . . Q . . |
| . Q . . . | | . . Q . . | | . . . . Q | | . . . Q . | | Q . . . . |
| . . . Q . | | . . . . Q | | . . Q . . | | Q . . . . | | . . . Q . |
| Q . . . . | | . Q . . . | | Q . . . . | | . . Q . . | | . Q . . . |
+-----------+ +-----------+ +-----------+ +-----------+ +-----------+
.
so a(2)=10.

Eduard I. Vatutin, Arranging of N queens on toroidal board and generating pandiagonal Latin squares using them (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).
Index entries for sequences related to Latin squares and rectangles.

Cf. A007705, A051906, A123565, A338562.

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

a(n) = A338562(n) / (2n)!. - Eduard I. Vatutin, Mar 13 2024

cp's OEIS Frontend

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

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A343866 Number of inequivalent cyclic diagonal Latin squares of order 2n+1 up to rotations, reflections and permutation of symbols.

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A366331 Number of main classes of diagonal Latin squares of order 2n+1 that contain a horizontally semicyclic Latin square.

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A370672 Number of ways of arranging 2n+1 nonattacking queens on a 2n+1 X 2n+1 toroidal board using knight moves.

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