A338620 Number of pandiagonal Latin squares of order 2n+1 with the first row in ascending order.

1, 0, 2, 4, 0, 8, 12386, 0

Offset: 0

Author

Eduard I. Vatutin, Nov 04 2020

Keywords

• nonn
• more
• hard

Comments

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

For orders n = 5, 7 and 11 all pandiagonal Latin squares are cyclic, so a(n) = A123565(2n+1) for n < 6. For n=6 (order 13), this is not true and there are 12386 inequivalent squares; of these 10 are cyclic (in all directions) and 1560 are semi-cyclic (A343867).

Pandiagonal Latin squares exist only for odd orders not divisible by 3. This is because the positions of each symbol are a solution to the toroidal n-queens problem which only has solutions for these sizes. - Andrew Howroyd, May 26 2021

Examples

			Example of a cyclic pandiagonal Latin square 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
Example of a cyclic pandiagonal Latin square of order 7:
  0 1 2 3 4 5 6
  2 3 4 5 6 0 1
  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
  5 6 0 1 2 3 4
Example of a cyclic pandiagonal Latin square of order 11:
   0  1  2  3  4  5  6  7  8  9 10
   2  3  4  5  6  7  8  9 10  0  1
   4  5  6  7  8  9 10  0  1  2  3
   6  7  8  9 10  0  1  2  3  4  5
   8  9 10  0  1  2  3  4  5  6  7
  10  0  1  2  3  4  5  6  7  8  9
   1  2  3  4  5  6  7  8  9 10  0
   3  4  5  6  7  8  9 10  0  1  2
   5  6  7  8  9 10  0  1  2  3  4
   7  8  9 10  0  1  2  3  4  5  6
   9 10  0  1  2  3  4  5  6  7  8
For order 13 there is a square
   7  1  0  3  6  5 12  2  8  9 10 11  4
   2  3  4 10  0  7  6  9 12 11  5  8  1
   4 11  1  7  8  9 10  3  6  0 12  2  5
   6  5  8 11 10  4  7  0  1  2  3  9 12
   8  9  2  5 12 11  1  4  3 10  0  6  7
   3  6 12  0  1  2  8 11  5  4  7 10  9
  10  0  3  2  9 12  5  6  7  8  1  4 11
   1  7 10  4  3  6  9  8  2  5 11 12  0
  11  4  5  6  7  0  3 10  9 12  2  1  8
   5  8  7  1  4 10 11 12  0  6  9  3  2
  12  2  9  8 11  1  0  7 10  3  4  5  6
   9 10 11 12  5  8  2  1  4  7  6  0  3
   0 12  6  9  2  3  4  5 11  1  8  7 10
that is pandiagonal but not cyclic (Dabbaghian and Wu).
		

Links

• A. O. L. Atkin, L. Hay, and R. G. Larson, Enumeration and construction of pandiagonal Latin squares of prime order, Computers & Mathematics with Applications, Volume. 9, Iss. 2, 1983, pp. 267-292.
• Vahid Dabbaghian and Tiankuang Wu, Constructing non-cyclic pandiagonal Latin squares of prime orders, Journal of Discrete Algorithms 30, 2015.
• Vahid Dabbaghian and Tiankuang Wu, Constructing Pandiagonal Latin Squares from Linear Cellular Automaton on Elementary Abelian Groups, Journal of Combinatorial Designs 23(5).

Crossrefs

Cf. A071607 (rows are cyclic), A123565, A342306, A343867 (semicyclic).

Formula

a(n) >= A123565(2n+1) + A343867(n). - Andrew Howroyd, May 26 2021

a(n) = A342306(n) / (2n+1)!. - Eduard I. Vatutin, Jun 13 2021

Extensions

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