A372922 Number of diagonal Latin squares of order 2n+1 that are isomorphic to cyclic Latin squares by row and column permutations.

1, 0, 480, 161280, 2229534720, 45984153600000, 3271798279766016000

Offset: 0

Author

Eduard I. Vatutin, May 16 2024

Keywords

• nonn
• more
• hard

Comments

The Latin squares considered here are diagonal Latin squares that are isomorphic to cyclic Latin squares. They are can be obtained from cyclic Latin squares (see A338522) by diagonalization (getting a corresponding pair of transversals and placing them on the diagonals, see article). These Latin squares have some interesting properties, for example, there are a large number of diagonal transversals.

Examples

			The cyclic Latin square of order 7
.
  0 1 2 3 4 5 6
  1 2 3 4 5 6 0
  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
  6 0 1 2 3 4 5
.
has a pair of symmetrically placed transversals T1 = (0, 2, 4, 6, 1, 3, 5) and T2 = (0, 5, 3, 1, 6, 4, 2), after permutting rown and columns transversal T1 placed to the main diagonal with getting single diagonal Latin square
.
  2 5 0 3 4 6 1
  0 3 5 1 2 4 6
  1 4 6 2 3 5 0
  6 2 4 0 1 3 5
  3 6 1 4 5 0 2
  4 0 2 5 6 1 3
  5 1 3 6 0 2 4
.
then after permuting rows and columns transversal T2 placed to the second diagonal with getting diagonal Latin square
.
  2 5 0 3 6 1 4
  0 3 5 1 4 6 2
  1 4 6 2 5 0 3
  6 2 4 0 3 5 1
  4 0 2 5 1 3 6
  5 1 3 6 2 4 0
  3 6 1 4 0 2 5
.
that can be canonized to the following diagonal Latin square:
.
  0 1 2 3 4 5 6
  2 3 1 5 6 4 0
  5 6 4 0 1 2 3
  4 0 6 2 3 1 5
  6 2 0 1 5 3 4
  1 5 3 4 0 6 2
  3 4 5 6 2 0 1
.
Cyclic Latin square of order 11
.
  0 1 2 3 4 5 6 7 8 9 10
  1 2 3 4 5 6 7 8 9 10 0
  2 3 4 5 6 7 8 9 10 0 1
  3 4 5 6 7 8 9 10 0 1 2
  4 5 6 7 8 9 10 0 1 2 3
  5 6 7 8 9 10 0 1 2 3 4
  6 7 8 9 10 0 1 2 3 4 5
  7 8 9 10 0 1 2 3 4 5 6
  8 9 10 0 1 2 3 4 5 6 7
  9 10 0 1 2 3 4 5 6 7 8
  10 0 1 2 3 4 5 6 7 8 9
.
can be diagonalized to set of diagonal Latin squares:
.
  0 1 2 3 4 5 6 7 8 9 10   0 1 2 3 4 5 6 7 8 9 10   0 1 2 3 4 5 6 7 8 9 10
  1 2 3 4 5 10 8 9 0 6 7   1 2 3 4 6 7 8 9 10 0 5   1 2 3 4 5 10 9 0 7 8 6
  3 4 5 10 7 9 1 8 2 0 6   8 10 5 7 9 3 0 4 1 6 2   3 4 5 10 6 9 7 2 1 0 8
  4 5 10 7 9 6 2 0 3 1 8   4 6 8 10 5 1 7 2 9 3 0   10 6 9 8 7 0 2 5 4 3 1
  10 7 9 6 8 0 4 2 5 3 1   9 0 1 2 3 10 4 5 6 7 8   9 8 7 0 1 2 4 6 10 5 3
  7 9 6 8 0 1 5 3 10 4 2   7 9 0 1 2 8 3 10 4 5 6   5 10 6 9 8 7 1 4 3 2 0
  8 0 1 2 3 4 9 10 6 7 5   6 8 10 5 7 2 9 3 0 4 1   7 0 1 2 3 4 10 8 9 6 5
  2 3 4 5 10 7 0 6 1 8 9   10 5 7 9 0 4 1 6 2 8 3   4 5 10 6 9 8 0 3 2 1 7
  5 10 7 9 6 8 3 1 4 2 0   3 4 6 8 10 0 5 1 7 2 9   8 7 0 1 2 3 5 9 6 10 4
  6 8 0 1 2 3 7 5 9 10 4   2 3 4 6 8 9 10 0 5 1 7   6 9 8 7 0 1 3 10 5 4 2
  9 6 8 0 1 2 10 4 7 5 3   5 7 9 0 1 6 2 8 3 10 4   2 3 4 5 10 6 8 1 0 7 9 ...
.
(totally 81 main classes of diagonal Latin squares).
		

Links

• 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.

Crossrefs

Cf. A338522, A338562, A342990, A372923, A375475.

Formula

a(n) = A372923(n) * (2n+1)!. - Eduard I. Vatutin, Sep 08 2024