1,1

A rook move on the equilateral triangle tessellation is a move along a path through successively edge-adjacent faces, that turns alternately left and right at each face (starting with either left or right), so that the overall direction of movement remains approximately parallel to one set of triangle edges.

Also counts the number of 3 X 2n matrices such that each row is a permutation of {1, .., 2n}, the first row is the identity permutation (1 .. 2n), and each column sums to either 3n+1 or 3n+2. This parallels how the equivalent problem for the board tessellated with hexagons (A002047) counts the number of 3 X (2n-1) zero-sum arrays.

For n = 2, the a(2) = 24 arrangements are rotations and reflections of:

o---o---o o---o---o o---o---o

/X\ / \ / \ /X\ / \ / \ /X\ / \ / \

o---o---o---o o---o---o---o o---o---o---o

/ \ / \ /X\ / \ / \ / \ / \X/ \ / \ / \ / \X/ \

o---o---o---o---o o---o---o---o---o o---o---o---o---o

\ / \ / \ / \X/ \ / \ / \ /X\ / \ /X\ / \ / \ /

o---o---o---o o---o---o---o o---o---o---o

\X/ \ / \ / \X/ \ / \ / \ / \ / \X/

o---o---o o---o---o o---o---o

(12 symmetries) (6 symmetries) (6 symmetries)

For n = 2, the a(2) = 24 matrices counted are:

1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4

2 3 4 1 2 3 4 1 2 4 1 3 2 4 3 1

4 2 1 3 4 3 1 2 4 2 3 1 4 1 2 3

-

1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4

2 4 3 1 3 1 4 2 3 2 4 1 3 2 4 1

4 2 1 3 3 4 1 2 3 4 1 2 4 3 1 2

-

1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4

3 4 1 2 3 4 1 2 3 4 2 1 3 4 2 1

4 1 3 2 4 2 3 1 4 1 2 3 4 1 3 2

plus the same matrices with rows 2 and 3 interchanged.

(MiniZinc)

% minizinc -D 'N=5' -s --all-solutions a375800.mzn

include "globals.mzn";

include "alldifferent.mzn";

int: N;

array[1..2*N] of var 1..2*N: perm1;

array[1..2*N] of var 1..2*N: perm2;

constraint forall(i in 1..2*N)(3*N+1 <= perm1[i]+perm2[i]+i /\ perm1[i]+perm2[i]+i <= 3*N+2);

constraint alldifferent(perm1);

constraint alldifferent(perm2);

solve satisfy;

output [show(i) ++ " " | i in 1..2*N];

output [show(perm1[i]) ++ " " | i in 1..2*N];

output [show(perm2[i]) ++ " " | i in 1..2*N];

Cf. A002047, A000170, A000142.

Sequence in context: A334775 A153389 A366000 * A332975 A010791 A145169

Adjacent sequences: A375797 A375798 A375799 * A375801 A375802 A375803

nonn,more

Hugh Robinson, Aug 29 2024

approved