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

0,3

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

Links

Table of n, a(n) for n=0..7.

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).

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

Example

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). (End)

Crossrefs

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

Sequence in context: A324717 A295793 A071607 * A358645 A095059 A021419

Adjacent sequences: A338617 A338618 A338619 * A338621 A338622 A338623

Keyword

nonn,more,hard,changed

Author

Eduard I. Vatutin, Nov 04 2020

Extensions

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

Status

approved