The On-Line Encyclopedia of Integer Sequences (OEIS)

A338522 revision #21

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