A341585 -id:A341585 - 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

A342998 Minimum number of diagonal transversals in a cyclic diagonal Latin square of order 2n+1.

Original entry on oeis.org

1, 0, 5, 27, 0, 4523, 128818, 0, 204330233, 11232045257

Offset: 0

Eduard I. Vatutin, Apr 02 2021

A cyclic Latin square is a Latin square in which row i is obtained by cyclically shifting row i-1 by d places (see A338562, A123565 and A341585).
Cyclic diagonal Latin squares do not exist for even orders.
a(n) <= A342997(n).
All cyclic diagonal Latin squares are diagonal Latin squares, so A287647(n) <= a((n-1)/2).

			For n=2 one of best cyclic diagonal Latin squares of order 5
  0 1 2 3 4
  2 3 4 0 1
  4 0 1 2 3
  1 2 3 4 0
  3 4 0 1 2
has a(2)=5 diagonal transversals:
  0 . . . .   . 1 . . .   . . 2 . .   . . . 3 .   . . . . 4
  . . 4 . .   . . . 0 .   . . . . 1   2 . . . .   . 3 . . .
  . . . . 3   4 . . . .   . 0 . . .   . . 1 . .   . . . 2 .
  . 2 . . .   . . 3 . .   . . . 4 .   . . . . 0   1 . . . .
  . . . 1 .   . . . . 2   3 . . . .   . 4 . . .   . . 0 . .

Eduard I. Vatutin, Enumerating the diagonal transversals for cyclic diagonal Latin squares of orders 1-19 (in Russian).
Eduard I. Vatutin, Proving list (best known examples).
Index entries for sequences related to Latin squares and rectangles.

Cf. A006717, A123565, A287647, A287648, A338562, A341585, A342997.

A342997 Maximum number of diagonal transversals in a cyclic diagonal Latin square of order 2n+1.

Original entry on oeis.org

1, 0, 5, 27, 0, 4665, 131106, 0, 204995269, 11254190082

Offset: 0

Eduard I. Vatutin, Apr 02 2021

A cyclic Latin square is a Latin square in which row i is obtained by cyclically shifting row i-1 by d places (see A338562, A123565 and A341585).
Cyclic diagonal Latin squares do not exist for even n.
All cyclic diagonal Latin squares are diagonal Latin squares, so a((n-1)/2) <= A287648(n).
All diagonal transversals are transversals, so a(n) <= A006717(n).
A342998 <= a(n).

			For n=2 one of the best cyclic diagonal Latin squares of order 5
  0 1 2 3 4
  2 3 4 0 1
  4 0 1 2 3
  1 2 3 4 0
  3 4 0 1 2
has a(2)=5 diagonal transversals:
  0 . . . .   . 1 . . .   . . 2 . .   . . . 3 .   . . . . 4
  . . 4 . .   . . . 0 .   . . . . 1   2 . . . .   . 3 . . .
  . . . . 3   4 . . . .   . 0 . . .   . . 1 . .   . . . 2 .
  . 2 . . .   . . 3 . .   . . . 4 .   . . . . 0   1 . . . .
  . . . 1 .   . . . . 2   3 . . . .   . 4 . . .   . . 0 . .

Eduard I. Vatutin, Enumerating the diagonal transversals for cyclic diagonal Latin squares of orders 1-19 (in Russian).
Eduard I. Vatutin, Proving list (best known examples).
Index entries for sequences related to Latin squares and rectangles.

Cf. A006717, A123565, A287648, A338562, A341585, A342998.

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.

A366333 a(n) is the number of distinct numbers of diagonal transversals that a semicyclic diagonal Latin square of order 2n+1 can have.

Original entry on oeis.org

1, 0, 1, 1, 0, 2, 20, 0, 271, 1208, 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). A vertically semicyclic diagonal Latin square is a square where each column c(i) is a cyclic shift of the first column c(0) by some value d(i). Cyclic diagonal Latin squares (see A338562) fall under the definition of vertically and horizontally semicyclic diagonal Latin squares simultaneously, in this type of squares each row r(i) is obtained from the previous one r(i-1) using cyclic shift by some value d.

Semicyclic diagonal Latin squares do not exist for even orders n.

			For n=6*2+1=13 the number of diagonal transversals that a semicyclic diagonal Latin square of order 13 may have is 127339, 127830, 128489, 128519, 128533, 128608, 128751, 128818, 128861, 129046, 129059, 129171, 129243, 129286, 129353, 129474, 129641, 129657, 130323 or 131106. Since there are 20 distinct values, a(6)=20.

Eduard I. Vatutin, About the spectra of numerical characteristics of different types of cyclic diagonal Latin squares (in Russian).
Eduard I. Vatutin, About the spectra of numerical characteristics of semicyclic diagonal Latin squares of order 17 (in Russian).
Eduard I. Vatutin, About the spectra of numerical characteristics of semicyclic diagonal Latin squares of order 19 (in Russian).
Eduard I. Vatutin, Proving lists (1, 5, 7, 11, 13, 17, 19).
Eduard I. Vatutin, Graphical representation of the spectra.
Index entries for sequences related to Latin squares and rectangles.

Cf. A071607, A341585, A342990, A345370, A366331.

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

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A342998 Minimum number of diagonal transversals in a cyclic diagonal Latin square of order 2n+1.

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A342997 Maximum number of diagonal transversals in a cyclic diagonal Latin square of order 2n+1.

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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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A366333 a(n) is the number of distinct numbers of diagonal transversals that a semicyclic diagonal Latin square of order 2n+1 can have.

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