A368027 Number of Dabbaghian-Wu pandiagonal Latin squares of order 2n+1.

1, 0, 0, 24, 0, 0, 72, 0, 0, 108, 0, 0, 4, 0, 0, 180, 0, 3, 216, 0, 0, 252, 0, 0, 264, 0, 0, 0, 0, 0, 360, 0, 5, 396, 0, 0, 432, 0, 0, 468, 0, 0, 0, 0, 0, 868, 0, 5, 576, 0

Offset: 1

Author

Eduard I. Vatutin, Dec 16 2023

Keywords

• nonn
• more

Comments

A pandiagonal Latin square is a Latin square in which the diagonal, antidiagonal and all broken diagonals and antidiagonals are transversals.

A Dabbaghian-Wu pandiagonal Latin square is a special type of pandiagonal Latin square (see A342306). Such squares are constructed from cyclic diagonal Latin squares (see A338562) for prime orders n=6k+1 (see Dabbaghian and Wu article) using a polynomial algorithm based on permutation of some values in Latin square. For other orders (25, 35, 49, ...) this algorithm also ensures correct pandiagonal Latin squares.

Examples

			n=13=6*2+1 (prime order):
.
  4 0 2 3 1 5 6 7 11 9 10 12 8
  11 12 1 10 6 2 4 5 3 7 8 9 0
  9 10 11 2 0 1 3 12 8 4 6 7 5
  6 8 9 7 11 12 0 4 2 3 5 1 10
  5 7 3 12 8 10 11 9 0 1 2 6 4
  3 4 8 6 7 9 5 1 10 12 0 11 2
  1 2 0 4 5 6 10 8 9 11 7 3 12
  0 9 5 1 3 4 2 6 7 8 12 10 11
  10 1 12 0 2 11 7 3 5 6 4 8 9
  8 6 10 11 12 3 1 2 4 0 9 5 7
  2 11 7 9 10 8 12 0 1 5 3 4 6
  7 5 6 8 4 0 9 11 12 10 1 2 3
  12 3 4 5 9 7 8 10 6 2 11 0 1
.
n=19=6*3+1 (prime order):
.
  8 0 2 3 4 6 17 7 1 9 10 11 12 13 14 15 16 5 18
  5 6 7 8 16 10 0 11 13 14 15 17 9 18 12 1 2 3 4
  10 4 12 13 14 15 16 17 18 0 8 2 11 3 5 6 7 9 1
  14 16 17 18 1 12 2 15 4 5 6 7 8 9 10 11 0 13 3
  1 2 3 11 5 14 6 8 9 10 12 4 13 7 15 16 17 18 0
  18 7 8 9 10 11 12 13 14 3 16 6 17 0 1 2 4 15 5
  11 12 13 15 7 16 10 18 0 1 2 3 4 5 6 14 8 17 9
  16 17 6 0 9 1 3 4 5 7 18 8 2 10 11 12 13 14 15
  2 3 4 5 6 7 8 9 17 11 1 12 14 15 16 18 10 0 13
  7 8 10 2 11 5 13 14 15 16 17 18 0 1 9 3 12 4 6
  12 1 14 4 15 17 18 0 2 13 3 16 5 6 7 8 9 10 11
  17 18 0 1 2 3 4 12 6 15 7 9 10 11 13 5 14 8 16
  3 5 16 6 0 8 9 10 11 12 13 14 15 4 17 7 18 1 2
  15 9 18 10 12 13 14 16 8 17 11 0 1 2 3 4 5 6 7
  13 14 15 16 17 18 7 1 10 2 4 5 6 8 0 9 3 11 12
  0 11 1 14 3 4 5 6 7 8 9 10 18 12 2 13 15 16 17
  4 13 5 7 8 9 11 3 12 6 14 15 16 17 18 0 1 2 10
  9 10 11 12 13 2 15 5 16 18 0 1 3 14 4 17 6 7 8
  6 15 9 17 18 0 1 2 3 4 5 13 7 16 8 10 11 12 14
.
n=25=6*4+1 (nonprime order):
.
  5 1 2 3 4 15 6 7 8 9 0 11 12 13 14 20 16 17 18 19 10 21 22 23 24
  3 4 20 6 7 8 9 15 11 12 13 14 5 16 17 18 19 0 21 22 23 24 10 1 2
  6 7 8 9 0 11 12 13 14 20 16 17 18 19 5 21 22 23 24 15 1 2 3 4 10
  9 15 11 12 13 14 0 16 17 18 19 10 21 22 23 24 5 1 2 3 4 20 6 7 8
  12 13 14 5 16 17 18 19 0 21 22 23 24 15 1 2 3 4 10 6 7 8 9 20 11
  20 16 17 18 19 10 21 22 23 24 5 1 2 3 4 15 6 7 8 9 0 11 12 13 14
  18 19 0 21 22 23 24 10 1 2 3 4 20 6 7 8 9 15 11 12 13 14 5 16 17
  21 22 23 24 15 1 2 3 4 10 6 7 8 9 0 11 12 13 14 20 16 17 18 19 5
  24 5 1 2 3 4 20 6 7 8 9 15 11 12 13 14 0 16 17 18 19 10 21 22 23
  2 3 4 10 6 7 8 9 20 11 12 13 14 5 16 17 18 19 0 21 22 23 24 15 1
  15 6 7 8 9 0 11 12 13 14 20 16 17 18 19 10 21 22 23 24 5 1 2 3 4
  8 9 15 11 12 13 14 5 16 17 18 19 0 21 22 23 24 10 1 2 3 4 20 6 7
  11 12 13 14 20 16 17 18 19 5 21 22 23 24 15 1 2 3 4 10 6 7 8 9 0
  14 0 16 17 18 19 10 21 22 23 24 5 1 2 3 4 20 6 7 8 9 15 11 12 13
  17 18 19 0 21 22 23 24 15 1 2 3 4 10 6 7 8 9 20 11 12 13 14 5 16
  10 21 22 23 24 5 1 2 3 4 15 6 7 8 9 0 11 12 13 14 20 16 17 18 19
  23 24 10 1 2 3 4 20 6 7 8 9 15 11 12 13 14 5 16 17 18 19 0 21 22
  1 2 3 4 10 6 7 8 9 0 11 12 13 14 20 16 17 18 19 5 21 22 23 24 15
  4 20 6 7 8 9 15 11 12 13 14 0 16 17 18 19 10 21 22 23 24 5 1 2 3
  7 8 9 20 11 12 13 14 5 16 17 18 19 0 21 22 23 24 15 1 2 3 4 10 6
  0 11 12 13 14 20 16 17 18 19 10 21 22 23 24 5 1 2 3 4 15 6 7 8 9
  13 14 5 16 17 18 19 0 21 22 23 24 10 1 2 3 4 20 6 7 8 9 15 11 12
  16 17 18 19 5 21 22 23 24 15 1 2 3 4 10 6 7 8 9 0 11 12 13 14 20
  19 10 21 22 23 24 5 1 2 3 4 20 6 7 8 9 15 11 12 13 14 0 16 17 18
  22 23 24 15 1 2 3 4 10 6 7 8 9 20 11 12 13 14 5 16 17 18 19 0 21
		

Links

• Vahid Dabbaghian and Tiankuang Wu, Constructing non-cyclic pandiagonal Latin squares of prime orders, Journal of Discrete Algorithms, Vol. 30, 2015, pp. 70-77, doi: 10.1016/j.jda.2014.12.001.
• Eduard I. Vatutin, About the Dabbaghian-Wu pandiagonal Latin squares for non-prime orders (in Russian).
• Index entries for sequences related to Latin squares and rectangles.

Crossrefs

Cf. A338562, A369379, A369380, A342306.