A338522 - OEIS

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

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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