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

1,2

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.

Links

Table of n, a(n) for n=1..21.

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).

Formula

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

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

Example

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.

Crossrefs

Cf. A000010, A000142, A074930, A123565.

Sequence in context: A029766 A088311 A359118 * A277183 A052588 A139239

Adjacent sequences: A338519 A338520 A338521 * A338523 A338524 A338525

Keyword

nonn,easy

Author

Eduard I. Vatutin, Nov 01 2020

Status

approved