A338522 Number of cyclic Latin squares of order n.

1, 2, 12, 48, 480, 1440, 30240, 161280, 2177280, 14515200, 399168000, 1916006400, 74724249600, 523069747200, 10461394944000, 167382319104000, 5690998849536000, 38414242234368000, 2189611807358976000, 19463216065413120000, 613091306060513280000

Offset: 1

Author

Eduard I. Vatutin, Nov 01 2020

Keywords

• nonn
• easy

Comments

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.

Examples

			For n=5 there are 4 cyclic Latin squares with the first row in natural order:
  0 1 2 3 4   0 1 2 3 4   0 1 2 3 4   0 1 2 3 4
  1 2 3 4 0   2 3 4 0 1   3 4 0 1 2   4 0 1 2 3
  2 3 4 0 1   4 0 1 2 3   1 2 3 4 0   3 4 0 1 2
  3 4 0 1 2   1 2 3 4 0   4 0 1 2 3   2 3 4 0 1
  4 0 1 2 3   3 4 0 1 2   2 3 4 0 1   1 2 3 4 0
and 4*5! = 480 cyclic Latin squares.
		

Links

• 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, About the number of cyclic Latin squares and cyclic diagonal Latin squares (in Russian).

Crossrefs

Cf. A000010, A000142, A074930, A123565.

Formula

a(n) = phi(n) * n!.

a(n) = A000010(n) * A000142(n).