A343868 Number of semicyclic Latin squares of order n with the first row in ascending order.

0, 0, 0, 8, 40, 338, 1512, 11368, 84960, 828972, 7291900, 85823668, 958954152, 12930529446, 176651211776, 2631044069296, 41847091313152

Offset

1,4

Comments

A semicyclic Latin square is cyclic in one or more directions but not in every direction. Cyclic Latin squares which are cyclic in essentially every direction are excluded. A direction here means any constant row and column displacement on the torus. See the reference in A343867 for additional information.

Each symbol occupies the same pattern of squares up to translation on the torus.

Links

Table of n, a(n) for n=1..17.

Andrew Howroyd, Order 6 semicyclic Latin squares

Andrew Howroyd, PARI Program

Formula

a(n) >= 2*((n-1)! - phi(n)).

a(p) = 2*(p-1)! + (p-1)*(A003111((p-1)/2) - p) for odd prime p.

Example

The permutation 164253 can be shown in a 6 X 6 grid:
    X . . . . .
    . . . . . X
    . . . X . .
    . X . . . .
    . . . . X .
    . . X . . .
This permutation gives the following 4 semicyclic squares.
    1 2 3 4 5 6   1 4 2 5 3 6   1 4 3 6 2 5   1 4 5 2 3 6
    2 3 4 5 6 1   2 5 3 6 4 1   3 6 2 5 4 1   2 5 6 3 4 1
    4 5 6 1 2 3   3 6 4 1 5 2   5 2 4 1 3 6   6 3 4 1 2 5
    6 1 2 3 4 5   4 1 5 2 6 3   4 1 6 3 5 2   4 1 2 5 6 3
    3 4 5 6 1 2   5 2 6 3 1 4   6 3 5 2 1 4   5 2 3 6 1 4
    5 6 1 2 3 4   6 3 1 4 2 5   2 5 1 4 6 3   3 6 1 4 5 2
In the third example, moving one cell down and two across increases the cell value by 1 (cyclically) and in the fourth example the displacement is 3 rows down and 2 across. Symbols can then be rearranged to give 4 distinct semicyclic squares with the first row in ascending order.

Prog

(PARI) \\ See Links

Crossrefs

Cf. A000010 (phi), A001615, A003111, A343867.

Sequence in context: A158922 A117083 A007987 * A350125 A096969 A376186

Adjacent sequences: A343865 A343866 A343867 * A343869 A343870 A343871

Keyword

nonn,more

Author

Andrew Howroyd, May 08 2021

Status

approved